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MTE 2: Linear Algebra

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Title Name	IGNOU MTE 2 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	MTE 2
Subject Name	Linear Algebra
Year	2026
Session	-
Language	English Medium
Assignment Code	MTE 2/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest MTE 02 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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MTE 2 2025 - English

Assignment

Course Code: MTE-02

Assignment Code:MTE-02/TMA/2025

Maximum Marks: 100

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

i) The function f: R→ R defined by f(x) = cosx is 1-1.

ii) The operation * defined by x + y = log(xy) is a binary operation on S, where S is the set {x ∈ R x > 0}.

The set {(x1.x2,...,xn) X1,X2,...,x ∈ R,X1 = 2x2+3} is a subspace of R".

iv) There is no 7 x 5 matrix of rank 6.

v) If V and V' are vector spaces and T: V → V' is a linear transformation, then whenever u1,u2,..., uk are linearly independent, Tu₁, Tu2, ..., Tuk are also linearly independent.

vi) If V is a vector space and T: V → V is a linear operator with det(T) = 0, then T is not diagonalisable.

vii) The degree of the minimal polynomial of a 3 x 3 matrix is at most 2.

viii) For any 2 x 2 matrix A, Adj (A') = (Adj(A))'.

ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.

x) There is no co-ordinate transformation that transforms the quadratic form x²+y² + z² to the quadratic form xz+yz.

2) a) Consider the funtion f: R{-1} → R defined by f(x) = 2x+1/ x+1

i) Check that f(x) is well defined and 1-1.

ii) Check that f(x) ≠ 2 for any x ∈ R.

iii) Check that g: R{2} → R given by g(x) = x-1/2-x is well defined and 1 -1. Further, check that g(x) -1 for any x ∈ R.

iv) Check that (fog)(x) = x for x ∈ R {2} and (gof) (x) = x for x ∈ R {-1}.

b) Find the direction cosines of the perpendicular from the origin to the plane r.(6i+4j+2√3k) +2 = 0.

3) Let V be the set of all functions that are twice differentiable in R and S = {cosx, sinx, xcosx,xsinx}.

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

a cosx + a₁ sinx+a2xcosx+a3x sinx.

Put x = 0, π,π/2,π/4, etc. and solve for a¡.)

b) Let W = [S] and let T: V → V be the function defined by

$T(f(x))=frac{d^2}{dx^2}(f(x))+2frac{d}{dx}(f(x))$

Check that T is a linear transformation on V.

c) Check that T(W) ⊂ W.

d) Write down the matrix of T on W w.r.t the basis S.

e) Is the matrix of the linear operator T non-singular? Justify your answer.

4) a) Show that, if A is any n x n matrix with real entries, then there is a n x n symmetric matrix S and a n x n skew symmetric matrix S' such that A = S+S'.

b) Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.

2a+3b+4c+d=8

a+2b+2c+2d = 3

a-b+c+3d = 3

a) For the following matrices, check whether there exists an invertible matrix P such that P-¹AP is diagonal. When such a P exists, find P.

i) $A=egin{bmatrix}0&1&-3\2&-1&6\1&-1&4end{bmatrix}$

ii) $B=egin{bmatrix}0&0&-2\1&2&1\1&0&3end{bmatrix}$

b) Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.

c) Using the fact that det(AB) = det(A) det(B) for any two matrices A and B, prove the identity

(a²+b²) (c2+d²) = (ac-bd)² + (ad+bc)²

a) Find the values of a, b ∈ C for which the matrix

$egin{bmatrix}1&i&1+i\a&0&b\1-i&2+i&1end{bmatrix}$

is Hermitian.

b) Are there values of a ∈ C for which the matrix

$egin{bmatrix}1&0&0\0&-1/sqrt{2}&1/sqrt{2}\0&1/sqrt{2}&aend{bmatrix}$

is unitary? Justify your answer.

c) Let (x₁,x₂,x₃) and (y₁, y₂, y₃) represent the coordinates with respect to the bases B₁ = {(1,0,0), (0,1,0), (0,0,1)}, B₂ = {(1,0,0), (0,1,2), (0,2,1)}. If Q(X)=x²+2x₁x₂+2x₂x₃+x2/2+x2/3, find the representation of Q in terms of (У₁, У₂, У_з).

a) Apply the Gram-Schmidt diagonalisation process to find an orthonormal basis for the subspace of C⁴ generated by the vectors

{(1,i,0,1), (1,0,i,0), (-i,0,1,-1)}

b) Find the orthogonal canonical reduction of the quadratic form x² - 2y² +z²+2xy+6yz and its principal axes. Also, find the rank and signature of the quadratic form.

MTE 2 2025 - Hindi

सत्रीय कार्य

पाठ्यक्रम कोड : मते-02

सत्रीय कार्य कोड: मते-02/त्मा/2025

अधिकतम अंक : 100

1. निम्नलिखित में से कौन से कथन सत्य हैं और कौन से असत्य हैं? लुघ उप्पत्ति या प्रति उदाहरण के साथ अपनी उत्तर की पुष्टि कीजिए।

i)f (x) = cos x द्वारा परिभाषित फलन $f:mathbf{R} ightarrowmathbf{R}, ext{1-1}$ है।

ii) x*y =log(xy) द्वारा परिभाषित संक्रिया S पर द्वि-आघारी संक्रिया है जहाँ S समुच्चय ${xinmathbf{R}mid x>0}$ है।

iii)समुच्चय ((x₁, x₂, ..., x_n) 1,x₁, x₂..., x_n∈ R,x₁ = 2x₂+3) R" की उपसमष्टि है।

iv) जाति 6 का कोई 7x5 आव्यूह नहीं होता।

v) यदि V और V सदिश समष्टियाँ हैं और $T:V ightarrow V'$ रैखिक रूपातंरण है, तब जब भी u₁, u₂,..., u_k रैखिकतः स्वतंत्र होते हैं, तब T_u1,T_u2,..., T_ukभी रैखिकतः स्वतंत्र होते हैं।

vi) यदि V एक सदिश समष्टि है और $T:V ightarrow Vquaddet(T)=0$ वाला रैखिक सँकारक है, तब T विकर्णनीय नहीं होता।

vii) एक 3x3 आव्यूह के अल्पिष्ठ बहुपद की कोटि अधिकतम 2 है।

viii) कोई भी 2x2 आव्यूह A के लिए $ext{Adj}(A')=( ext{Adj}(A))'$

ix) केवल शून्य आव्यूह ऐसा आव्यूह है जो सममित और विषम सममित दोनों होता है।

x) कोई भी ऐसा निर्देशांक रूपांतरण नहीं है जो द्विघाती समघात x² + y² +z² को द्विघाती समघात xz + yz में रूपांतरित करता है।

2. क) $f(x)=frac{2x+1}{x+1}$ द्वारा परिभाषित फलन फ: $f:mathbf{R}setminus{-1} ightarrowmathbf{R}$ लीजिएः

i) जाँच कीजिए कि फf(x) सुपरिभाषित और 1-1 है।

ii) जाँच कीजिए कि किसी x∈ Rके लिए f(x) #2 है।

iii) जाँच कीजिए कि $g(x)=frac{x-1}{2-x}$ द्वारा दिया गया f: R{-1} $o$ R सुपरिभाषित और 1-1 है। इसके आगे जाँच कीजिए कि किसी x∈R के लिए g (x)# -1 है।

iv) जाँच कीजिए कि x∈ R{2} के लिए (fog) (x) = x और x∈ R {-1} के लिए (fog) (x) = x

ख) मूल बिन्दु से समतल र. $mathbf{r}cdot(6mathbf{i}+4mathbf{j}+2sqrt{3}mathbf{k})+2=0$ के लंब की दिक्कोज्याएं ज्ञात कीजिए।

3. मान लीजिए व ऐसे सभी फलनो का समुच्चय है जो R में दो बार अवकलनीय हैं और ऍस = $S={cos x,sin x,xcos x,xsin x}$

क) जाँच कीजिए किS.R पर रैखिकतः स्वतंत्र समुच्चय है। (संकेतः समीकरण $a_0cos x+a_1sin x+a_2 xcos x+a_3 xsin xquad ext{}$

$x=0,pi,frac{pi}{2},frac{pi}{4}$ इत्यादि रखिए और a₁ के हल कीजिए)

ख) मान लीजिए W = [S] और मान लीजिए $T:V ightarrow V,quad Tf(x)=frac{d^2}{dx^2}(f(x))+2frac{d}{dx}(f(x))$ द्वारा परिभाषित फलन है। जाँच कीजिए कि T,V पर रैखिक रूपांतरण है।

ग) जाँच कीजिए T(W)⊂ W

घ) आधार Sके सापेक्ष W पर T का आव्यूह लिखिए।

ड़) क्या रैखिक संकारक T का आव्यूह व्युत्क्रमणीय है। अपने उत्तर की पुष्टि कीजिए।

4.क) दिखाइए कि यदि A वास्तविक प्रविष्टियों वाला nXn आव्यूह है तब एक ऐसा mXn सममित आव्यूह S और nXn विषम सममित आव्यूह S' हैं जिसके लिए A= S + S'.

ख) पंक्ति बराबर चरण रूप में संगत मैट्रिक्स को बराबर करके समीकरणों के निम्नलिखित निकायों के समाधान खोजें:

2a + 3b + 4c + d = 8
a + 2b + 2c + 2d = 3
a - b + c + 3d = 3

5. क) निम्नलिखित आव्यूहों के लिए, जाँच कीजिए कि ऐसे व्युत्क्रमणीय आव्यूह प का अस्तित्व होता है जिसके लिए P^-1AP विकर्ण है। जब ऐसे P का अस्तित्व होता है तब P ज्ञात कीजिए।

$ext{i)}A=egin{bmatrix}0&1&-3\2&-1&6\1&-1&4end{bmatrix}qquad ext{ii)}B=egin{bmatrix}0&0&-3\1&2&1\1&0&3end{bmatrix}$

ख) कैली-हैमिल्टन प्रमेय द्वारा भाग क) में आव्यूह B का व्युत्क्रम ज्ञात कीजिए।

ग) किन्ही दो आव्यूहों A और B के date (AB) = date(A) देट (B) तथ्य का प्रयोग करके निम्नलिखित समिका सिद्ध कीजिए।

$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$

6. क) ऐसे a b∈ C के मान ज्ञात कीजिए जिसके लिए आव्यूह

$egin{bmatrix}1&i&1+i\a&0&b\1-i&2+i&1end{bmatrix}$

हर्मिटी है।

ख) क्या ऐसेa∈cC क के मान होते हैं जिसके लिए आव्यूह $egin{bmatrix}1&0&0\0&-1/sqrt{2}&1/sqrt{2}\0&1/sqrt{2}&aend{bmatrix}$

ऐकिक है? अपने उत्त्तर की पुष्टि कीजिए।

ग) मान लिजिए (x₁, x₂, x₃) और (Y_¹, y₂ y₃) आधारो B₁ = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}, B₂ = {(1, 0, 0), (0, 1, 2), (0, 2, 1)) के सापेक्ष निर्देशांको को निरूपित करते है। यदि Q(x) =x^₂_1+x²+₂X,X₂ + ₂X₂x + x₂ + x₃, तब (y₁, y₂ y₃) के पदों में क्यु का निरूपण ज्ञात कीजिए।

7.क) सदिशों {(1, 1, 0, 1), (1, 0, 1, 0), (-1, 0, 1, -1)} द्वारा जनित की उपसमष्टि के लिए प्रसामान्य लांबिक आधार ज्ञात करने के लिए ग्राम-रिमट लांबिकीकरण प्रक्रम लागू कीजिए।

ख) द्विधाती समघात x²-2y² + 2²+2xy+6yz का लांबिक विहित समानयन और इसके मुख्य अक्ष ज्ञात कीजिए। द्विधाती समघात की जाति और चिन्ह भी ज्ञात कीजिए।

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