IGNOU BMTE 141 SOLVED ASSIGNMENT HINDI

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IGNOU BMTE 141 SOLVED ASSIGNMENT HINDI

~~Rs. 90~~
Rs. 15

IGNOU BMTE 141 SOLVED ASSIGNMENT HINDI

~~Rs. 90~~
Rs. 15

Last Date of Submission of IGNOU BMTE-141 (BAG) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
Semester Wise
January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam).
July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

Title Name	IGNOU BMTE 141 SOLVED ASSIGNMENT HINDI
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BAG
Course Name	BACHELOR OF ARTS
Subject Code	BMTE 141
Subject Name	Linear Algebra
Year	2025
Session
Language	English Medium
Assignment Code	BMTE-141/Assignmentt-1//2025
Product Description	Assignment of BAG (BACHELOR OF ARTS) 2025. Latest BMTE 141 2025 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BMTE-141 (BAG) 2025 Assignment is for January 2025 Session: 30th September, 2025 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2025 (for June 2025 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).

~~Rs. 90~~
Rs. 15

Questions Included in this Help Book

Ques 1.

Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k.

Ques 2.

Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1,1,0).

Ques 3.

Check whether W = {(x, y, z) ∈ R³ |x + y - z = 0} is a subspace of R³.

Ques 4.

Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.

Ques 5.

Check whether T: R² → R², defined by T(x, y) = (-y, x) is a linear transformation.

Ques 6.

If {U¹, U²} is an ordered basis of R2 and {f_₁ (v), f₂ (v)} is the corresponding dual basis find f₁ (2v₁+v₂) and f₂ (U₁-2v₂).

Ques 7.

Find the kernel of the linear transformation T: R² → R² defined by T(x, y) = (2x + 3y, 2x – 3y).

Ques 8.

Describe the linear transformation T:R² → R² such that

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

where B is the standard basis of R².

Ques 9.

Find the matrix of the linear transformation T: R² → R² defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.

Ques 10.

Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + C^tD defined? Justify your answer.

Ques 11.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 12.

Check whether $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 13.

Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product

$\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$

Find the inner product of the functions f(t) = 2t,g(t) = 1/ t²+5.

Ques 14.

Find adjoint of the linear operator T: C² → C² defined by T (z₁, z₂) = (z₂, z₁ + iz₂) with respect to the standard inner product on C².

Ques 15.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 16.

Let S be any non-empty set and let V(S) be the set of all real valued functions on R. Define addition on V(s) by (f+g)(x) = f(x) + g(x) and scalar multiplication by (af)(x) = af (x). Check that (V(S), +, -) is a vector space.

Ques 17.

Check that B = {1, 2x + 1, (x - 1)²} is a basis for P_₂, the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 18.

Let T: R³ → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$ is $\begin{bmatrix}1&0&-1\\0&1&1\\1&0&1\end{bmatrix}$ Find the matrix of the linear transformation with respect to the basis

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 19.

Show that W = {(x, 4x, 3x) ∈ R²x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R³.

Ques 20.

Find the eigenvalues and eigenvectors of the matrix B = $\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ . Is the matrix diagonalisable? Justify your answer.

Ques 21.

Find Adj(A) where A = $\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ . Hence find A^-1.

Ques 22.

Solve the folowing set of simultaneous equations using Cramer's rule:

x+2y+ z = 3

2x - y+2z = 1

3x+y+z=0

Ques 23.

Find the minimal polynomial of the matrix

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 24.

Let V be the vector space of all real valued functions that are twice differentiable in R and

S = {cos x, sin. x, x cos x, x sin x}.

Check that S is a linearly independent set over R. (Hint: Consider the equation

a ₀cos x + a_₁ sin x + a₂x cos x + a₃x sin x.

(Put x = 0, π, π/2,π/4, etc. and find a₁.)

Ques 25.

Consider the linear operator T: C³ → C³, defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3)$

i) Compute T* and check whether Tis self-adjoint.

ii) Check whether T is unitary.

Ques 26.

Let (x₁, x₂, x₃) and (y₁, y₂, y₃) represent the coordinates with respect to the bases B₁ = {(1,0,0), (1, 1, 0), (0, 0, 1)}, B₂ = {(1,0,0), (0, 1, 1), (0, 0, 1)}. If $Q(X)=x_{1}^{2}-4x_{1}x_{2}+2x_{2}x_{3}+x_{2}^{2}+x_{3}^{2}$ find the representation of Q in terms of (y₁, y₂, y₃).

Ques 27.

Find the orthogonal canonical reduction of the quadratic form-x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes.

Ques 28.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

Ques 29.

सदिशों $\sqrt{2}i+2j+2k$ और $i+\sqrt{2}j+\sqrt{2}k$ के बीच का कोण ज्ञात कीजिये।.

Ques 30.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 31.

जाँच कीजिये कि $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}\mathbb{R}^3$ की उप सम्ष्टि है या नहीं।

Ques 32.

जाँच कीजिये कि $p^3$ में, जो कोटि 3 य उस से कम वाले बहुपदों की सदिश स्मष्टि है, सदिशों की शमष्टि $\{1+x,x+x^2,1+x^3\}$ रैखिकतः स्वतंत्र है।.

Ques 33.

जाँच कीजिये कि $T:\mathbb{R}^2\rightarrow\mathbb{R}^2,\text{}T(x,y)=(-y,x)$ द्वारा परिभाषित है, एक रैखिक रूपांतरण है।

Ques 34.

यदि $\{v_1,v_2\}\in\mathbb{R}^2$ का एक क्रमित आधार है और $\{f_1(v),f_2(v)\}$ इस का संगत द्वैत आधार है तो $f_1(2v_1+v_2)$ और $f_2(v_1-2v_2)$ निकालिये।

Ques 35.

खिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ की अष्टि ज्ञात कीजिये, जो ? (?, ?) = (2? + 3?, 2? − 3?) द्वारा परिभाषित है,।

Ques 36.

उस रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ को वर्णन कीजिये जिस् के लिये

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

है जहाँ $B\mathbb{R}^2$ के मानक आधार है।

Ques 37.

रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ के क्रमित आधार {(0, −1), (−1, 0)} के सापेक्ष आव्यूह निकालिये, जो ? (?, ?) = (2?, ? − ?) द्वारा परिभाषित है,।.

Ques 38.

मान लीजिये कि A एक 2 x 3 आव्युह है, B एक 3 x 4 आव्यूह है, C एक 3 x 2 आव्यूह है और D एक 3 x 4 आव्यूह है। क्या AB + C^tD परिभाषित है? अपने उत्तर की पुष्टि कीजिये।

Ques 39.

आव्यूह $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$ के लिये कैली-हैल्टन प्रमेय सत्यापित कीजिये।

Ques 40.

जाँच कीजिये कि $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ आव्यूह के $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ आइगेंसदिश है। संगत आईगेमान क्या है?

Ques 41.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

के सापेक्ष वास्तविक मूल्यों वाले उत्पादों का प्रतिच्छेदन गुणक यौगिक है। $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}$ आंतर गुणन का गुणनफल ज्ञात कीजिए।

Ques 42.

रैखिक संकारक ? ∶ ℂ² → ℂ², जो ? (z₁,z₂) = (z₂,z₁ + iz₂) द्वारा परिभाषित है, का C² पर मानक अंतर गुणन के सापेक्ष संलग्न निकालिये।

Ques 43.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 44.

मान लीजिये कि S कोई भी एक अरिक्त समुच्चय है और V (S), सपर सभी वास्तविक मान वाले फलनों का समुच्चय है। V(S) पर योग (f+g)(x) = f(x) + g(x) द्वारा परिभषित कीजिये और आदिश गुणन ( $\alpha$ f) (x) = $\alpha$ f(x) द्वारा परिभषित कीजिये। जाँच कीजिये कि V (S), +, .) एक सदिश समष्टि है।

Ques 45.

जाँच कीजिये कि B = {1, 2? + 1, (? − 1)2 }, ?₂ के लिये एक आधार है जहाँ पी2 अधिक से अधिक कोटि P² वाले वास्तविक गुणाँक बहुपदों की सदिश समष्टि है।

Ques 46.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 47.

दिखाइये कि $W=\{(x,4x,3x)\in\mathbb{R}^3\mid x\in\mathbb{R}\}\mathbb{R}^3$ की एक उपसमष्टि है। $\mathbb{R}^3$ की उपस्मष्टि U का आधार भी ज्ञात कीजिये, जो $W\oplus U=\mathbb{R}^3$ , को संतुष्ट करती है।

Ques 48.

आव्यूह $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\\\end{bmatrix}$ के आइगेमान और आइगेंसदिश ज्ञात कीजिये। क्या यह आव्यूह विकर्णनीय अपने उत्तर की पुष्टि कीजिये।

Ques 49.

ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

Ques 50.

निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 51.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 52.

मान लीजिये कि व, आईआर पर ऐसे सभी वास्तविक मान वाले फलनों की सदिश समष्टि है जो दो बार अवकलनीय है। जाँच कीजिए कि ऍस पर रैखिकतः स्वतंत्र है। (संकेतः समीकरण

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 53.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 54.

मान लीजिये कि (?₁ , ?₂ , ?₃) और (y₁, y₂, y₃) आधारों ?₁ = {(1, 0, 0), (1, 1, 0), (0, 0, 1)},, ?₂ = {(1, 0, 0), (0, 1, 1), (0, 0, 1)}, के राापेक्ष निर्देशांकों को निरुपित करते हैं। यदि

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 55.

द्विघाति समघात −?2 + ?2 + ?2 + 4?? + 4?z का लाम्बिक विहित समानयन और इस के मुक्य अक्ष निकालिये।

Ques 56.

निम्न लिखित कथनों में से कौन-से कथन सत्य और कौन-से कथन असत्य? अपने उत्तर की एक लघु उपपत्ति या प्रति-उदाहरण द्वारा पुष्टि कीजिये।

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

ii) यदि V एक सदिश सम्ष्टि है और $S=\{v_1,v_2,\dots,v_n\}\subset V,\quad n\ge 3,$ एक ऐसा समुच्चय हैजिस में $v_i\ne v_j$ यदि $i\neq j$ , तो ऍस एक रैखिकतः स्वतंत्र समुच्चय है।

iii) यदि ?₁ , ?₂ ∶ ? → ? एक परिमित विमा ? और ?1 ∘ ?2 व्युत्क्रमणीय हैं, तो ?2 ∘ ?1 भी वव्युत्क्रमणीय है।

iv) यदि एक ? × ?, ? ≥ 2,, वर्गीय आव्यूह विकर्णनीय है, तो इस के अल्पिष्ट बहुपद और अभिलक्षणिक बहुपद बराबर है।

v) यदि ?1 , ?2 ∶ ? → ? एक परिमित विमा सदिश समष्टि ? पर स्वसंलग्न संकारक है, तो T₁ + T₂ भी स्वसंलग्न है।

Ques 57.

Find the angle between the vectors $\sqrt{2}\mathbf{i}+2\mathbf{j}+2\mathbf{k}\text{and}\mathbf{i}+\sqrt{2}\mathbf{j}+\sqrt{2}\mathbf{k}\text{is}0^\circ.$

Ques 58.

Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0)

Ques 59.

Check whether $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}$ is a subspace of ℝ³

Ques 60.

Check whether the set of vectors $\{1+x,x+x^2,1+x^3\}$ is a linearly independent set of

vectors in P_{3 ,}the vector space of polynomials of degree ≤ 3.

Ques 61.

Check whetherr ? ∶ ℝ² → ℝ², defined by ? (?, ?) = (−?, ?) is a linear transformation.

Ques 62.

If {?1 , ?2 } is an ordered basis of ℝ ²and {?1 (?) , ?2 (?)} is the corresponding dual basis find

$f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 63.

Find the kernel of the linear transformation ? ∶ ℝ² → ℝ² defined by $f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 64.

Describe the linear transformation ? ∶ ℝ²→ ℝ² such that $[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$ where ? is the standard basis of ℝ²

Ques 65.

Find the matrix of the linear transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2\text{defined by}T(x,y)=(2y,x-y)$ with respect to the ordered basis{(0, −1), (−1, 0)}.

Ques 66.

Let ? be a 2 × 3 matrix, ? be a 3 × 4 matrix and ? be a 3 × 2 matrix and ? be a 3 × 4 matrix. Is ?? + ??? defined? Justify your answer.

Ques 67.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 68.

Check whethe $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 69.

Let ?[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner produc

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Find the inner product of the functions $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}.$

Ques 70.

Find adjoint of the linear operator ? ∶ ℂ² → ℂ² defined by $T(z_1,z_2)=(z_2,z_1+iz_2)$ respect to the standard inner product on ℂ 2 .

Ques 71.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 72.

Let ? be any non-empty set and let ? (?) be the set of all real valued functions on ℝ. Define addition on $(f+g)(x)=f(x)+g(x)$ and scalar multiplication by $(\alpha\cdot f)(x)=\alpha f(x).$ Check that $(V(S),+,\cdot)$ is a vector space.

Ques 73.

Check that $B=\{1,2x+1,(x-1)^2\}$ is a basis for ?², the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 74.

Let ? ∶ ℝ³ → ℝ³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

$\begin{bmatrix}1&0&-1\\0&1&1\\1&0&1\end{bmatrix}$ . Find the matrix of the linear transformation with respect to the basis $B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 75.

Find the eigenvalues and eigenvectors of the matrix $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ Is the matrix diagonalisable? Justify your answer.

Ques 76.

Solve the folowing set of simultaneous equations using Cramer’s rule:

$x+2y+z=3$

$2x-y+2z=1$

3? + ? + ? = 0

Ques 77.

Find the minimal polynomial of the matrix $\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 78.

Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and $S=\{\cos x,\sin x,x\cos x,x\sin x\}.$

Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation $a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x.$

$\left(\text{Put}x=0,\pi,\frac{\pi}{2},\frac{\pi}{4},\text{etc.and find}a_i.\right)$

Ques 79.

Consider the linear operator ? ∶ ℂ³ → ℂ³ , defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3).$

i) Compute ? ∗ and check whether ? is self-adjoint

ii) Check whether ? is unitary.

Ques 80.

$\text{Let}(x_1,x_2,x_3)\text{and}(y_1,y_2,y_3)$ represent the coordinates with respect to the bases

$B_1=((1,0,0),(1,1,0),(0,0,1)),\quad B_2=((1,0,0),(0,1,1),(0,0,1)).$

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

find the representation of ? in terms of (?1 , ?2 , ?3).

Ques 81.

Find the orthogonal canonical reduction of the quadratic form $-x^2+y^2+z^2+4xy+4xz.$ Also, find its principal axes.

Ques 82.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample

i) If ?1 and ?₂ are proper subspaces of a non-zero, finite dimensional, vector space ? and

$\dim(W_1)>\frac{\dim(V)}{2},\dim(W_2)>\frac{\dim(V)}{2},\text{then}W_1\cap W_2\neq\{0\}.$

ii) If ? is a vector space and $S=\{v_1,v_2,\dots,v_n\}\subset V,n\ge 3,$ is such that ?? ≠ ?? if ? ≠ ?, then S

is a linearly independent set

iii) $\text{If}T_1,T_2:V\rightarrow V\text{are}$ ? are linear operators on a finite dimensional vector space ? and ?₁ ∘ ?₂ is invertible, ?₂ ∘ ?₁is also invertibl

iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial

v) If ?₁ , ?₂ ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?₁ + ?₂ is also a self adjoint operator.

Ques 83.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 84.

जाँच कीजिये कि में, जो कोटि 3 य उस से कम वाले बहुपदों की सदिश स्मष्टि है, सदिशों की शमष्टि $\{1+x,x+x^2,1+x^3\}$ रैखिकतः स्वतंत्र है।.

Ques 85.

Ques 86.

यदि का एक क्रमित आधार है और $\{f_1(v),f_2(v)\}$ इस का संगत द्वैत आधार है तो $f_1(2v_1+v_2)$ और $f_2(v_1-2v_2)$ निकालिये।

Ques 87.

रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ की अष्टि ज्ञात कीजिये, जो ? (?, ?) = (2? + 3?, 2? − 3?) द्वारा परिभाषित है,।

Ques 88.

Ques 89.

Ques 90.

Ques 91.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

के सापेक्ष वास्तविक मूल्यों वाले उत्पादों का प्रतिच्छेदन गुणक यौगिक है। आंतर गुणन का गुणनफल ज्ञात कीजिए।

Ques 92.

Ques 93.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 94.

Ques 95.

Ques 96.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 97.

3) a) आव्यूह $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\\\end{bmatrix}$ के आइगेमान और आइगेंसदिश ज्ञात कीजिये। क्या यह आव्यूह विकर्णनीय अपने उत्तर की पुष्टि कीजिये।

Ques 98.

Adj(A) ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

4) a) निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 99.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 100.

1) a) मान लीजिये कि व, आईआर पर ऐसे सभी वास्तविक मान वाले फलनों की सदिश समष्टि है जो दो बार अवकलनीय है। जाँच कीजिए कि ऍस पर रैखिकतः स्वतंत्र है। (संकेतः समीकरण

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 101.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 102.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 103.

Ques 104.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 105.

यदि V एक सदिश सम्ष्टि है और $S=\{v_1,v_2,\dots,v_n\}\subset V,\quad n\ge 3,$ एक ऐसा समुच्चय हैजिस में $v_i\ne v_j$ यदि $i\neq j$ , तो ऍस एक रैखिकतः स्वतंत्र समुच्चय है।

Ques 106.

यदि ?₁ , ?₂ ∶ ? → ? एक परिमित विमा ? और ?1 ∘ ?2 व्युत्क्रमणीय हैं, तो ?2 ∘ ?1 भी वव्युत्क्रमणीय है।

Ques 107.

यदि एक ? × ?, ? ≥ 2,, वर्गीय आव्यूह विकर्णनीय है, तो इस के अल्पिष्ट बहुपद और अभिलक्षणिक बहुपद बराबर है।

Ques 108.

यदि ?1 , ?2 ∶ ? → ? एक परिमित विमा सदिश समष्टि ? पर स्वसंलग्न संकारक है, तो T₁ + T₂ भी स्वसंलग्न है।

Ques 109.

Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k.

Ques 110.

Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1,1,0).

Ques 111.

Check whether W = {(x, y, z) ∈ R³ |x + y - z = 0} is a subspace of R³.

Ques 112.

Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.

Ques 113.

Check whether T: R² → R², defined by T(x, y) = (-y, x) is a linear transformation.

Ques 114.

If {U¹, U²} is an ordered basis of R2 and {f_₁ (v), f₂ (v)} is the corresponding dual basis find f₁ (2v₁+v₂) and f₂ (U₁-2v₂).

Ques 115.

Find the kernel of the linear transformation T: R² → R² defined by T(x, y) = (2x + 3y, 2x – 3y).

Ques 116.

Describe the linear transformation T:R² → R² such that

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

where B is the standard basis of R².

Ques 117.

Find the matrix of the linear transformation T: R² → R² defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.

Ques 118.

Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + C^tD defined? Justify your answer.

Ques 119.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 120.

Check whether $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 121.

Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product

$\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$

Find the inner product of the functions f(t) = 2t,g(t) = 1/ t²+5.

Ques 122.

Find adjoint of the linear operator T: C² → C² defined by T (z₁, z₂) = (z₂, z₁ + iz₂) with respect to the standard inner product on C².

Ques 123.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 124.

Ques 125.

Check that B = {1, 2x + 1, (x - 1)²} is a basis for P_₂, the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 126.

Let T: R³ → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 127.

Show that W = {(x, 4x, 3x) ∈ R²x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R³.

Ques 128.

Find the eigenvalues and eigenvectors of the matrix B = $\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ . Is the matrix diagonalisable? Justify your answer.

Ques 129.

Find Adj(A) where A = $\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ . Hence find A^-1.

Ques 130.

Solve the folowing set of simultaneous equations using Cramer's rule:

x+2y+ z = 3

2x - y+2z = 1

3x+y+z=0

Ques 131.

Find the minimal polynomial of the matrix

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 132.

Let V be the vector space of all real valued functions that are twice differentiable in R and

S = {cos x, sin. x, x cos x, x sin x}.

Check that S is a linearly independent set over R. (Hint: Consider the equation

a ₀cos x + a_₁ sin x + a₂x cos x + a₃x sin x.

(Put x = 0, π, π/2,π/4, etc. and find a₁.)

Ques 133.

Consider the linear operator T: C³ → C³, defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3)$

i) Compute T* and check whether Tis self-adjoint.

ii) Check whether T is unitary.

Ques 134.

Ques 135.

Find the orthogonal canonical reduction of the quadratic form-x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes.

Ques 136.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

Ques 137.

सदिशों $\sqrt{2}i+2j+2k$ और $i+\sqrt{2}j+\sqrt{2}k$ के बीच का कोण ज्ञात कीजिये।.

Ques 138.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 139.

जाँच कीजिये कि $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}\mathbb{R}^3$ की उप सम्ष्टि है या नहीं।

Ques 140.

Ques 141.

Ques 142.

Ques 143.

Ques 144.

उस रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ को वर्णन कीजिये जिस् के लिये

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

है जहाँ $B\mathbb{R}^2$ के मानक आधार है।

Ques 145.

Ques 146.

Ques 147.

आव्यूह $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$ के लिये कैली-हैल्टन प्रमेय सत्यापित कीजिये।

Ques 148.

Ques 149.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Ques 150.

Ques 151.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 152.

Ques 153.

Ques 154.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 155.

Ques 156.

Ques 157.

ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

Ques 158.

निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 159.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 160.

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 161.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 162.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 163.

Ques 164.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 165.

Find the angle between the vectors $\sqrt{2}\mathbf{i}+2\mathbf{j}+2\mathbf{k}\text{and}\mathbf{i}+\sqrt{2}\mathbf{j}+\sqrt{2}\mathbf{k}\text{is}0^\circ.$

Ques 166.

Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0)

Ques 167.

Check whether $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}$ is a subspace of ℝ³

Ques 168.

Check whether the set of vectors $\{1+x,x+x^2,1+x^3\}$ is a linearly independent set of

vectors in P_{3 ,}the vector space of polynomials of degree ≤ 3.

Ques 169.

Check whetherr ? ∶ ℝ² → ℝ², defined by ? (?, ?) = (−?, ?) is a linear transformation.

Ques 170.

If {?1 , ?2 } is an ordered basis of ℝ ²and {?1 (?) , ?2 (?)} is the corresponding dual basis find

$f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 171.

Find the kernel of the linear transformation ? ∶ ℝ² → ℝ² defined by $f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 172.

Describe the linear transformation ? ∶ ℝ²→ ℝ² such that $[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$ where ? is the standard basis of ℝ²

Ques 173.

Find the matrix of the linear transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2\text{defined by}T(x,y)=(2y,x-y)$ with respect to the ordered basis{(0, −1), (−1, 0)}.

Ques 174.

Let ? be a 2 × 3 matrix, ? be a 3 × 4 matrix and ? be a 3 × 2 matrix and ? be a 3 × 4 matrix. Is ?? + ??? defined? Justify your answer.

Ques 175.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 176.

Check whethe $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 177.

Let ?[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner produc

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Find the inner product of the functions $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}.$

Ques 178.

Find adjoint of the linear operator ? ∶ ℂ² → ℂ² defined by $T(z_1,z_2)=(z_2,z_1+iz_2)$ respect to the standard inner product on ℂ 2 .

Ques 179.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 180.

Ques 181.

Check that $B=\{1,2x+1,(x-1)^2\}$ is a basis for ?², the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 182.

Let ? ∶ ℝ³ → ℝ³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 183.

Find the eigenvalues and eigenvectors of the matrix $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ Is the matrix diagonalisable? Justify your answer.

Ques 184.

Solve the folowing set of simultaneous equations using Cramer’s rule:

$x+2y+z=3$

$2x-y+2z=1$

3? + ? + ? = 0

Ques 185.

Find the minimal polynomial of the matrix $\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 186.

Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and $S=\{\cos x,\sin x,x\cos x,x\sin x\}.$

Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation $a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x.$

$\left(\text{Put}x=0,\pi,\frac{\pi}{2},\frac{\pi}{4},\text{etc.and find}a_i.\right)$

Ques 187.

Consider the linear operator ? ∶ ℂ³ → ℂ³ , defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3).$

i) Compute ? ∗ and check whether ? is self-adjoint

ii) Check whether ? is unitary.

Ques 188.

$\text{Let}(x_1,x_2,x_3)\text{and}(y_1,y_2,y_3)$ represent the coordinates with respect to the bases

$B_1=((1,0,0),(1,1,0),(0,0,1)),\quad B_2=((1,0,0),(0,1,1),(0,0,1)).$

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

find the representation of ? in terms of (?1 , ?2 , ?3).

Ques 189.

Find the orthogonal canonical reduction of the quadratic form $-x^2+y^2+z^2+4xy+4xz.$ Also, find its principal axes.

Ques 190.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample

i) If ?1 and ?₂ are proper subspaces of a non-zero, finite dimensional, vector space ? and

$\dim(W_1)>\frac{\dim(V)}{2},\dim(W_2)>\frac{\dim(V)}{2},\text{then}W_1\cap W_2\neq\{0\}.$

ii) If ? is a vector space and $S=\{v_1,v_2,\dots,v_n\}\subset V,n\ge 3,$ is such that ?? ≠ ?? if ? ≠ ?, then S

is a linearly independent set

iii) $\text{If}T_1,T_2:V\rightarrow V\text{are}$ ? are linear operators on a finite dimensional vector space ? and ?₁ ∘ ?₂ is invertible, ?₂ ∘ ?₁is also invertibl

iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial

v) If ?₁ , ?₂ ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?₁ + ?₂ is also a self adjoint operator.

Ques 191.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 192.

Ques 193.

Ques 194.

Ques 195.

Ques 196.

Ques 197.

Ques 198.

Ques 199.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Ques 200.

Ques 201.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 202.

Ques 203.

Ques 204.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 205.

Ques 206.

Adj(A) ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

4) a) निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 207.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 208.

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 209.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 210.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 211.

Ques 212.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 213.

Ques 214.

Ques 215.

Ques 216.

Ques 217.

Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k.

Ques 218.

Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1,1,0).

Ques 219.

Check whether W = {(x, y, z) ∈ R³ |x + y - z = 0} is a subspace of R³.

Ques 220.

Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.

Ques 221.

Check whether T: R² → R², defined by T(x, y) = (-y, x) is a linear transformation.

Ques 222.

If {U¹, U²} is an ordered basis of R2 and {f_₁ (v), f₂ (v)} is the corresponding dual basis find f₁ (2v₁+v₂) and f₂ (U₁-2v₂).

Ques 223.

Find the kernel of the linear transformation T: R² → R² defined by T(x, y) = (2x + 3y, 2x – 3y).

Ques 224.

Describe the linear transformation T:R² → R² such that

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

where B is the standard basis of R².

Ques 225.

Find the matrix of the linear transformation T: R² → R² defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.

Ques 226.

Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + C^tD defined? Justify your answer.

Ques 227.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 228.

Check whether $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 229.

Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product

$\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$

Find the inner product of the functions f(t) = 2t,g(t) = 1/ t²+5.

Ques 230.

Find adjoint of the linear operator T: C² → C² defined by T (z₁, z₂) = (z₂, z₁ + iz₂) with respect to the standard inner product on C².

Ques 231.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 232.

Ques 233.

Check that B = {1, 2x + 1, (x - 1)²} is a basis for P_₂, the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 234.

Let T: R³ → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 235.

Show that W = {(x, 4x, 3x) ∈ R²x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R³.

Ques 236.

Find the eigenvalues and eigenvectors of the matrix B = $\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ . Is the matrix diagonalisable? Justify your answer.

Ques 237.

Find Adj(A) where A = $\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ . Hence find A^-1.

Ques 238.

Solve the folowing set of simultaneous equations using Cramer's rule:

x+2y+ z = 3

2x - y+2z = 1

3x+y+z=0

Ques 239.

Find the minimal polynomial of the matrix

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 240.

Let V be the vector space of all real valued functions that are twice differentiable in R and

S = {cos x, sin. x, x cos x, x sin x}.

Check that S is a linearly independent set over R. (Hint: Consider the equation

a ₀cos x + a_₁ sin x + a₂x cos x + a₃x sin x.

(Put x = 0, π, π/2,π/4, etc. and find a₁.)

Ques 241.

Consider the linear operator T: C³ → C³, defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3)$

i) Compute T* and check whether Tis self-adjoint.

ii) Check whether T is unitary.

Ques 242.

Ques 243.

Find the orthogonal canonical reduction of the quadratic form-x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes.

Ques 244.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

Ques 245.

सदिशों $\sqrt{2}i+2j+2k$ और $i+\sqrt{2}j+\sqrt{2}k$ के बीच का कोण ज्ञात कीजिये।.

Ques 246.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 247.

जाँच कीजिये कि $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}\mathbb{R}^3$ की उप सम्ष्टि है या नहीं।

Ques 248.

Ques 249.

Ques 250.

Ques 251.

Ques 252.

उस रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ को वर्णन कीजिये जिस् के लिये

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

है जहाँ $B\mathbb{R}^2$ के मानक आधार है।

Ques 253.

Ques 254.

Ques 255.

आव्यूह $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$ के लिये कैली-हैल्टन प्रमेय सत्यापित कीजिये।

Ques 256.

Ques 257.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Ques 258.

Ques 259.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 260.

Ques 261.

Ques 262.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 263.

Ques 264.

Ques 265.

ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

Ques 266.

निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 267.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 268.

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 269.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 270.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 271.

Ques 272.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 273.

Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k.

Ques 274.

Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1,1,0).

Ques 275.

Check whether W = {(x, y, z) ∈ R³ |x + y - z = 0} is a subspace of R³.

Ques 276.

Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.

Ques 277.

Check whether T: R² → R², defined by T(x, y) = (-y, x) is a linear transformation.

Ques 278.

If {U¹, U²} is an ordered basis of R2 and {f_₁ (v), f₂ (v)} is the corresponding dual basis find f₁ (2v₁+v₂) and f₂ (U₁-2v₂).

Ques 279.

Find the kernel of the linear transformation T: R² → R² defined by T(x, y) = (2x + 3y, 2x – 3y).

Ques 280.

Describe the linear transformation T:R² → R² such that

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

where B is the standard basis of R².

Ques 281.

Find the matrix of the linear transformation T: R² → R² defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.

Ques 282.

Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + C^tD defined? Justify your answer.

Ques 283.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 284.

Check whether $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 285.

Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product

$\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$

Find the inner product of the functions f(t) = 2t,g(t) = 1/ t²+5.

Ques 286.

Find adjoint of the linear operator T: C² → C² defined by T (z₁, z₂) = (z₂, z₁ + iz₂) with respect to the standard inner product on C².

Ques 287.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 288.

Ques 289.

Check that B = {1, 2x + 1, (x - 1)²} is a basis for P_₂, the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 290.

Let T: R³ → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 291.

Show that W = {(x, 4x, 3x) ∈ R²x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R³.

Ques 292.

Find the eigenvalues and eigenvectors of the matrix B = $\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ . Is the matrix diagonalisable? Justify your answer.

Ques 293.

Find Adj(A) where A = $\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ . Hence find A^-1.

Ques 294.

Solve the folowing set of simultaneous equations using Cramer's rule:

x+2y+ z = 3

2x - y+2z = 1

3x+y+z=0

Ques 295.

Find the minimal polynomial of the matrix

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 296.

Let V be the vector space of all real valued functions that are twice differentiable in R and

S = {cos x, sin. x, x cos x, x sin x}.

Check that S is a linearly independent set over R. (Hint: Consider the equation

a ₀cos x + a_₁ sin x + a₂x cos x + a₃x sin x.

(Put x = 0, π, π/2,π/4, etc. and find a₁.)

Ques 297.

Consider the linear operator T: C³ → C³, defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3)$

i) Compute T* and check whether Tis self-adjoint.

ii) Check whether T is unitary.

Ques 298.

Ques 299.

Find the orthogonal canonical reduction of the quadratic form-x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes.

Ques 300.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

Ques 301.

सदिशों $\sqrt{2}i+2j+2k$ और $i+\sqrt{2}j+\sqrt{2}k$ के बीच का कोण ज्ञात कीजिये।.

Ques 302.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 303.

जाँच कीजिये कि $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}\mathbb{R}^3$ की उप सम्ष्टि है या नहीं।

Ques 304.

Ques 305.

Ques 306.

Ques 307.

Ques 308.

उस रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ को वर्णन कीजिये जिस् के लिये

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

है जहाँ $B\mathbb{R}^2$ के मानक आधार है।

Ques 309.

Ques 310.

Ques 311.

आव्यूह $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$ के लिये कैली-हैल्टन प्रमेय सत्यापित कीजिये।

Ques 312.

Ques 313.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Ques 314.

Ques 315.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 316.

Ques 317.

Ques 318.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 319.

Ques 320.

Ques 321.

ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

Ques 322.

निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 323.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 324.

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 325.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 326.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 327.

Ques 328.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 329.

Find the angle between the vectors $\sqrt{2}\mathbf{i}+2\mathbf{j}+2\mathbf{k}\text{and}\mathbf{i}+\sqrt{2}\mathbf{j}+\sqrt{2}\mathbf{k}\text{is}0^\circ.$

Ques 330.

Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0)

Ques 331.

Check whether $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}$ is a subspace of ℝ³

Ques 332.

Check whether the set of vectors $\{1+x,x+x^2,1+x^3\}$ is a linearly independent set of

vectors in P_{3 ,}the vector space of polynomials of degree ≤ 3.

Ques 333.

Check whetherr ? ∶ ℝ² → ℝ², defined by ? (?, ?) = (−?, ?) is a linear transformation.

Ques 334.

If {?1 , ?2 } is an ordered basis of ℝ ²and {?1 (?) , ?2 (?)} is the corresponding dual basis find

$f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 335.

Find the kernel of the linear transformation ? ∶ ℝ² → ℝ² defined by $f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 336.

Describe the linear transformation ? ∶ ℝ²→ ℝ² such that $[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$ where ? is the standard basis of ℝ²

Ques 337.

Find the matrix of the linear transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2\text{defined by}T(x,y)=(2y,x-y)$ with respect to the ordered basis{(0, −1), (−1, 0)}.

Ques 338.

Let ? be a 2 × 3 matrix, ? be a 3 × 4 matrix and ? be a 3 × 2 matrix and ? be a 3 × 4 matrix. Is ?? + ??? defined? Justify your answer.

Ques 339.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 340.

Check whethe $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 341.

Let ?[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner produc

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Find the inner product of the functions $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}.$

Ques 342.

Find adjoint of the linear operator ? ∶ ℂ² → ℂ² defined by $T(z_1,z_2)=(z_2,z_1+iz_2)$ respect to the standard inner product on ℂ 2 .

Ques 343.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 344.

Ques 345.

Check that $B=\{1,2x+1,(x-1)^2\}$ is a basis for ?², the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 346.

Let ? ∶ ℝ³ → ℝ³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 347.

Find the eigenvalues and eigenvectors of the matrix $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ Is the matrix diagonalisable? Justify your answer.

Ques 348.

Solve the folowing set of simultaneous equations using Cramer’s rule:

$x+2y+z=3$

$2x-y+2z=1$

3? + ? + ? = 0

Ques 349.

Find the minimal polynomial of the matrix $\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 350.

Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and $S=\{\cos x,\sin x,x\cos x,x\sin x\}.$

Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation $a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x.$

$\left(\text{Put}x=0,\pi,\frac{\pi}{2},\frac{\pi}{4},\text{etc.and find}a_i.\right)$

Ques 351.

Consider the linear operator ? ∶ ℂ³ → ℂ³ , defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3).$

i) Compute ? ∗ and check whether ? is self-adjoint

ii) Check whether ? is unitary.

Ques 352.

$\text{Let}(x_1,x_2,x_3)\text{and}(y_1,y_2,y_3)$ represent the coordinates with respect to the bases

$B_1=((1,0,0),(1,1,0),(0,0,1)),\quad B_2=((1,0,0),(0,1,1),(0,0,1)).$

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

find the representation of ? in terms of (?1 , ?2 , ?3).

Ques 353.

Find the orthogonal canonical reduction of the quadratic form $-x^2+y^2+z^2+4xy+4xz.$ Also, find its principal axes.

Ques 354.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample

i) If ?1 and ?₂ are proper subspaces of a non-zero, finite dimensional, vector space ? and

$\dim(W_1)>\frac{\dim(V)}{2},\dim(W_2)>\frac{\dim(V)}{2},\text{then}W_1\cap W_2\neq\{0\}.$

ii) If ? is a vector space and $S=\{v_1,v_2,\dots,v_n\}\subset V,n\ge 3,$ is such that ?? ≠ ?? if ? ≠ ?, then S

is a linearly independent set

iii) $\text{If}T_1,T_2:V\rightarrow V\text{are}$ ? are linear operators on a finite dimensional vector space ? and ?₁ ∘ ?₂ is invertible, ?₂ ∘ ?₁is also invertibl

iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial

v) If ?₁ , ?₂ ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?₁ + ?₂ is also a self adjoint operator.

Ques 355.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 356.

Ques 357.

Ques 358.

Ques 359.

Ques 360.

Ques 361.

Ques 362.

Ques 363.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Ques 364.

Ques 365.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 366.

Ques 367.

Ques 368.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 369.

Ques 370.

Adj(A) ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

4) a) निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 371.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 372.

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 373.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 374.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 375.

Ques 376.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 377.

Ques 378.

Ques 379.

Ques 380.

Ques 381.

Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k.

Ques 382.

Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1,1,0).

Ques 383.

Check whether W = {(x, y, z) ∈ R³ |x + y - z = 0} is a subspace of R³.

Ques 384.

Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.

Ques 385.

Check whether T: R² → R², defined by T(x, y) = (-y, x) is a linear transformation.

Ques 386.

If {U¹, U²} is an ordered basis of R2 and {f_₁ (v), f₂ (v)} is the corresponding dual basis find f₁ (2v₁+v₂) and f₂ (U₁-2v₂).

Ques 387.

Find the kernel of the linear transformation T: R² → R² defined by T(x, y) = (2x + 3y, 2x – 3y).

Ques 388.

Describe the linear transformation T:R² → R² such that

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

where B is the standard basis of R².

Ques 389.

Find the matrix of the linear transformation T: R² → R² defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.

Ques 390.

Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + C^tD defined? Justify your answer.

Ques 391.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 392.

Check whether $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 393.

Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product

$\langle f,g\rangle=\int_{0}^{1}f(t)g(t)dt$

Find the inner product of the functions f(t) = 2t,g(t) = 1/ t²+5.

Ques 394.

Find adjoint of the linear operator T: C² → C² defined by T (z₁, z₂) = (z₂, z₁ + iz₂) with respect to the standard inner product on C².

Ques 395.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 396.

Ques 397.

Check that B = {1, 2x + 1, (x - 1)²} is a basis for P_₂, the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 398.

Let T: R³ → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 399.

Show that W = {(x, 4x, 3x) ∈ R²x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R³.

Ques 400.

Find the eigenvalues and eigenvectors of the matrix B = $\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ . Is the matrix diagonalisable? Justify your answer.

Ques 401.

Find Adj(A) where A = $\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ . Hence find A^-1.

Ques 402.

Solve the folowing set of simultaneous equations using Cramer's rule:

x+2y+ z = 3

2x - y+2z = 1

3x+y+z=0

Ques 403.

Find the minimal polynomial of the matrix

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 404.

Let V be the vector space of all real valued functions that are twice differentiable in R and

S = {cos x, sin. x, x cos x, x sin x}.

Check that S is a linearly independent set over R. (Hint: Consider the equation

a ₀cos x + a_₁ sin x + a₂x cos x + a₃x sin x.

(Put x = 0, π, π/2,π/4, etc. and find a₁.)

Ques 405.

Consider the linear operator T: C³ → C³, defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3)$

i) Compute T* and check whether Tis self-adjoint.

ii) Check whether T is unitary.

Ques 406.

Ques 407.

Find the orthogonal canonical reduction of the quadratic form-x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes.

Ques 408.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

Ques 409.

सदिशों $\sqrt{2}i+2j+2k$ और $i+\sqrt{2}j+\sqrt{2}k$ के बीच का कोण ज्ञात कीजिये।.

Ques 410.

बिंदुओं (1,0,-1), (0, 1, 1) और (-1,1,0) द्वारा निर्धारित समतल की सदिश समीकरण ज्ञात कीजिये ।.

Ques 411.

जाँच कीजिये कि $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}\mathbb{R}^3$ की उप सम्ष्टि है या नहीं।

Ques 412.

Ques 413.

Ques 414.

Ques 415.

Ques 416.

उस रैखिक रूपांतरण $T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ को वर्णन कीजिये जिस् के लिये

$[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$

है जहाँ $B\mathbb{R}^2$ के मानक आधार है।

Ques 417.

Ques 418.

Ques 419.

आव्यूह $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$ के लिये कैली-हैल्टन प्रमेय सत्यापित कीजिये।

Ques 420.

Ques 421.

मान लीजिये कि C[0, 1] अंतराल [0, 1] पर आंतर गुणन फलन

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Ques 422.

Ques 423.

द्विघाति समघात $x_1^2-2x_2^2+3x_3^2$ का चिह्नक इकालिये।

Ques 424.

Ques 425.

Ques 426.

मान लीजिये कि ? ∶ ℝ³ → ℝ³ एक रैखिक संकारक है ओर क्रमित आधार

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष उसका आव्यूह है। क्रमित आधार

$B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

के सापेक्ष T का आव्यूह निकालिये।।

Ques 427.

Ques 428.

Ques 429.

ज्ञात कीजिये, जहाँ $A=\begin{bmatrix}3&2&2\\-1&1&0\\3&0&1\end{bmatrix}$ इस से A^-1 निकालिये।

Ques 430.

निम्नलिखित समीकरण निकाय को क्रमर नियम से हल कीजिये:

$\begin{aligned}x+2y+z&=3\\2x-y+2z&=1\\3x+y+z&=0\end{aligned}$

Ques 431.

आव्यूह

$\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

की अल्पिष्ठ बहुपद ज्ञात कीजिये।

Ques 432.

$a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x=0$

लीजिये। $x=0,\pi,\frac{\pi}{2},\frac{\pi}{4}$ इत्यादि रखिये और a_i निकालिये।)

Ques 433.

रैखिक समीकरण टीसीसी³ लीजिये जो

? (?₁ , ?₂ , ?₃) = (?₁ − ??₂ , ??₁ − 2?₂ + ??₃ , −??₂ + ?₃) .

द्वारा परिभषित है।

i) T* परिकलित कीजिये और जाँच कीजिये कि ट स्वसंलग्न है।

ii) जाँच कीजिये कि ट ऐकिक है।

Ques 434.

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

(?1 , ?2 , ?3) के पदों में Q को निरूपण निकालिये।

Ques 435.

Ques 436.

i) यदि W₁ और W₂ एक परिमित विमा, शून्येत्तर सदिश समष्टि V की उपसमष्टियाँ हैं और

$\dim(W_1)>\frac{\dim(V)}{2},\quad\dim(W_2)>\frac{\dim(V)}{2},\quad\text{}W_1\cap W_2\ne\{0\}.$

Ques 437.

Find the angle between the vectors $\sqrt{2}\mathbf{i}+2\mathbf{j}+2\mathbf{k}\text{and}\mathbf{i}+\sqrt{2}\mathbf{j}+\sqrt{2}\mathbf{k}\text{is}0^\circ.$

Ques 438.

Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (−1, 1, 0)

Ques 439.

Check whether $W=\{(x,y,z)\in\mathbb{R}^3\mid x+y-z=0\}$ is a subspace of ℝ³

Ques 440.

Check whether the set of vectors $\{1+x,x+x^2,1+x^3\}$ is a linearly independent set of

vectors in P_{3 ,}the vector space of polynomials of degree ≤ 3.

Ques 441.

Check whetherr ? ∶ ℝ² → ℝ², defined by ? (?, ?) = (−?, ?) is a linear transformation.

Ques 442.

If {?1 , ?2 } is an ordered basis of ℝ ²and {?1 (?) , ?2 (?)} is the corresponding dual basis find

$f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 443.

Find the kernel of the linear transformation ? ∶ ℝ² → ℝ² defined by $f_1(2v_1+v_2)\text{and}f_2(v_1-2v_2).$

Ques 444.

Describe the linear transformation ? ∶ ℝ²→ ℝ² such that $[T]_B=\begin{bmatrix}1&2\\2&0\end{bmatrix}$ where ? is the standard basis of ℝ²

Ques 445.

Find the matrix of the linear transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2\text{defined by}T(x,y)=(2y,x-y)$ with respect to the ordered basis{(0, −1), (−1, 0)}.

Ques 446.

Let ? be a 2 × 3 matrix, ? be a 3 × 4 matrix and ? be a 3 × 2 matrix and ? be a 3 × 4 matrix. Is ?? + ??? defined? Justify your answer.

Ques 447.

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

Ques 448.

Check whethe $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

Ques 449.

Let ?[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner produc

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Find the inner product of the functions $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}.$

Ques 450.

Find adjoint of the linear operator ? ∶ ℂ² → ℂ² defined by $T(z_1,z_2)=(z_2,z_1+iz_2)$ respect to the standard inner product on ℂ 2 .

Ques 451.

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

Ques 452.

Ques 453.

Check that $B=\{1,2x+1,(x-1)^2\}$ is a basis for ?², the vector space of polynomials with real coefficients of degree ≤ 2.

Ques 454.

Let ? ∶ ℝ³ → ℝ³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

Ques 455.

Find the eigenvalues and eigenvectors of the matrix $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ Is the matrix diagonalisable? Justify your answer.

Ques 456.

Solve the folowing set of simultaneous equations using Cramer’s rule:

$x+2y+z=3$

$2x-y+2z=1$

3? + ? + ? = 0

Ques 457.

Find the minimal polynomial of the matrix $\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

Ques 458.

Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and $S=\{\cos x,\sin x,x\cos x,x\sin x\}.$

Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation $a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x.$

$\left(\text{Put}x=0,\pi,\frac{\pi}{2},\frac{\pi}{4},\text{etc.and find}a_i.\right)$

Ques 459.

Consider the linear operator ? ∶ ℂ³ → ℂ³ , defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3).$

i) Compute ? ∗ and check whether ? is self-adjoint

ii) Check whether ? is unitary.

Ques 460.

$\text{Let}(x_1,x_2,x_3)\text{and}(y_1,y_2,y_3)$ represent the coordinates with respect to the bases

$B_1=((1,0,0),(1,1,0),(0,0,1)),\quad B_2=((1,0,0),(0,1,1),(0,0,1)).$

$Q(X)=x_1^2-4x_1x_2+2x_2x_3+x_2^2+x_3^2$

find the representation of ? in terms of (?1 , ?2 , ?3).

Ques 461.

Find the orthogonal canonical reduction of the quadratic form $-x^2+y^2+z^2+4xy+4xz.$ Also, find its principal axes.

Ques 462.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample

i) If ?1 and ?₂ are proper subspaces of a non-zero, finite dimensional, vector space ? and

$\dim(W_1)>\frac{\dim(V)}{2},\dim(W_2)>\frac{\dim(V)}{2},\text{then}W_1\cap W_2\neq\{0\}.$

ii) If ? is a vector space and $S=\{v_1,v_2,\dots,v_n\}\subset V,n\ge 3,$ is such that ?? ≠ ?? if ? ≠ ?, then S

is a linearly independent set

iii) $\text{If}T_1,T_2:V\rightarrow V\text{are}$ ? are linear operators on a finite dimensional vector space ? and ?₁ ∘ ?₂ is invertible, ?₂ ∘ ?₁is also invertibl

iv) If an ? × ? square matrix, ? ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial

v) If ?₁ , ?₂ ∶ ? → ? are self adjoint operators on a finite dimensional inner product space ?, then ?₁ + ?₂ is also a self adjoint operator.

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