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MTE-02 Solved Assignment January 2026 | रैखिक बीजगणित (Linear Algebra) | IGNOU B.SC.

Q1. 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. (100 words)
ii)) The operation * defined by x* y = log(xy) is a binary operation on S, where S is the set {x ∈ R|x > 0}. (100 words)
iii)) The set {(x1.x2,...,xn) | X1,X2,...,xn ∈ R, x1 = 2x2 + 3} is a subspace of Rn. (100 words)
iv)) There is no 7 × 5 matrix of rank 6. (100 words)
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. (100 words)
vi)) If V is a vector space and T : V → V is a linear operator with det(T) = 0, then T is not diagonalisable. (100 words)
vii)) The degree of the minimal polynomial of a 3 × 3 matrix is at most 2. (100 words)
viii)) For any 2 × 2 matrix A, Adj (A') = (Adj(A))'. (100 words)
ix)) The only matrix which is both symmetric and skew-symmetric is the zero matrix. (100 words)
x)) There is no co-ordinate transformation that transforms the quadratic form x² + y² + z² to the quadratic form xz + yz. (100 words)

Key Points:

Cosine function is not 1-1 due to its periodic nature.
Binary operations require closure: results must stay within the set.
Subspaces must contain the zero vector; otherwise, they are not subspaces.
Matrix rank is limited by `min(rows, columns)`; a `7x5` matrix cannot have rank 6.

Answer: This document comprehensively addresses ten statements related to linear algebra concepts from the MTE-02 course. For each statement, its truthfulness is assessed, followed by a detailed justification involving either a short proof or a counterexample. The explanations are designed to align with typical IGNOU course content, covering fundamental definitions, properties of functions, vector spaces, matrices, linear transformations, and quadratic forms. Each sub-question is treated independently ...

Q2. N/A

a)) Consider the funtion f : R\{-1} → R defined by f(x) = 2x+1 / x+1. (650 words)
a)i)) Check that f(x) is well defined and 1 – 1. (150 words)
a)ii)) Check that f(x) ≠ 2 for any x ∈ R. (100 words)
a)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. (200 words)
a)iv)) Check that (f°g)(x) = x for x ∈ R \ {2} and (g ◦ f)(x) = x for x ∈ R \{-1}. (200 words)
b)) Find the direction cosines of the perpendicular from the origin to the plane r. (6i+4j+2√3k) + 2 = 0. (100 words)

Key Points:

A function is well-defined if its denominator is never zero in its domain.
A function f is 1-1 if f(x₁) = f(x₂) implies x₁ = x₂.
To check range exclusion, assume f(x)=c and show it leads to a contradiction.
Function composition (f°g)(x) means applying g first, then f to the result.

Answer: This answer addresses the properties of functions, including well-definedness, injectivity (1-1 mapping), range exclusions, and function composition. It also covers the calculation of direction cosines for a plane's normal vector. The problem involves algebraic manipulation and understanding fundamental definitions from Linear Algebra, particularly in the context of functions and vector geometry.

Q3. 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 a0 cosx + a1 sinx + a2xcosx+a3x sinx. Put x = 0, π/2, π, 3π/2, etc. and solve for ai.) (250 words)
b)) Let W = [S] and let T : V → V be the function defined by T(f(x)) = d²/dx² (f(x))+2d/dx (f(x)). Check that T is a linear transformation on V. (150 words)
c)) Check that T(W) ⊂W. (350 words)
d)) Write down the matrix of T on W w.r.t the basis S. (100 words)
e)) Is the matrix of the linear operator T non-singular? Justify your answer. (150 words)

Key Points:

Linear independence requires trivial linear combination of basis vectors.
A linear transformation preserves vector addition and scalar multiplication.
Subspace invariance means T maps elements of W back into W.
Matrix of T has images of basis vectors as its columns.

Answer: This response comprehensively addresses the properties of a given set of functions within the vector space of twice-differentiable functions, focusing on linear independence, verification of a linear transformation, invariance of a subspace, and the matrix representation and singularity of the transformation. Concepts from MTE-02 Linear Algebra, such as vector space properties, linear transformations, and matrix determinants, are applied throughout the analysis. Each sub-question is answered met...

Q4. N/A

a)) Show that, if A is any n × n matrix with real entries, then there is a n × n symmetric matrix S and a n × n skew symmetric matrix S' such that A = S+S'. (150 words)
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 (250 words)

Key Points:

Any square matrix A can be uniquely decomposed into symmetric (S) and skew-symmetric (S') parts.
A symmetric matrix S satisfies Sᵀ = S.
A skew-symmetric matrix S' satisfies (S')ᵀ = -S'.
The symmetric component is S = (1/2)(A + Aᵀ), and the skew-symmetric component is S' = (1/2)(A - Aᵀ).

Answer:

Q5. N/A

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=[[0,1,-3],[2,-1,6],[1,-1,4]] ii) B=[[0,0,-2],[1,2,1],[1,0,3]] (550 words)
b)) Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem. (150 words)
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. (150 words)

Key Points:

A matrix is diagonalizable if and only if for each eigenvalue, its algebraic multiplicity equals its geometric multiplicity.
Geometric multiplicity of an eigenvalue is the dimension of its eigenspace, found by solving (A - λI)v = 0.
The Cayley-Hamilton theorem states a square matrix satisfies its own characteristic polynomial.
Cayley-Hamilton theorem can be used to find a matrix inverse: B⁻¹ = (-1/c₀)(Bⁿ⁻¹ + ... + c₁I).

Answer: This response addresses the given questions from MTE-02 Linear Algebra, focusing on matrix diagonalization, inverse calculation using the Cayley-Hamilton theorem, and a proof involving determinants. It utilizes fundamental concepts such as eigenvalues, eigenvectors, algebraic and geometric multiplicities, and properties of matrix multiplication and determinants, which are core topics within the IGNOU MTE-02 curriculum. For diagonalization, the existence of an invertible matrix P depends on whet...

Q6. N/A

a)) Find the values of a, b ∈ C for which the matrix [[1, i, a],[0, b, 1-i],[1+i, 2+i, 1]] is Hermitian. (100 words)
b)) Are there values of a ∈ C for which the matrix [[0, 0, 1],[1/√2, 1/√2, 0],[1/√2, 0, a]] is unitary ? Justify your answer. (150 words)
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) = x1²+2x1x2+2x2x3+x2²+x3², find the representation of Q in terms of (y1,y2,y3). (150 words)

Key Points:

A matrix A is Hermitian if A = A* (conjugate transpose), where A_ij = (A_ji)*.
Diagonal elements of a Hermitian matrix must always be real numbers.
A matrix U is unitary if U*U = I, meaning its row/column vectors form an orthonormal set.
Orthonormal vectors must have a magnitude of 1 and be mutually orthogonal (dot product is zero).

Answer: This response comprehensively addresses the properties of Hermitian and unitary matrices, and the transformation of quadratic forms under a change of basis, drawing upon fundamental concepts from Linear Algebra typically covered in MTE-02. It provides step-by-step calculations and justifications for each sub-question. For a matrix to be Hermitian, its conjugate transpose must be equal to itself. For a matrix to be unitary, its inverse must be equal to its conjugate transpose, which implies its ...

Q7. N/A

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)}. (300 words)
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. (300 words)

Key Points:

Gram-Schmidt uses complex inner product: <u, v> = uᵀv̄ to orthogonalize vectors in Cⁿ.
Orthogonalization involves subtracting projections of subsequent vectors onto previously orthogonalized vectors.
Normalization ensures vectors have unit length: u = v / ||v||, where ||v|| = √<v, v>.
Quadratic form Q(x) = xᵀAx is reduced by finding eigenvalues (λ) and eigenvectors of symmetric matrix A.

Answer:

MTE-02 Solved Assignment — January 2026 / July 2026

रैखिक बीजगणित (Linear Algebra) | IGNOU Bachelor of Science (B.SC.)

MTE-02—January 2026 / July 2026

रैखिक बीजगणित (Linear Algebra)

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MTE-02 Assignment Questions — January 2026

Q1. Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

Q2. N/A

Q3. Let V be the set of all functions that are twice differentiable in R and S = {cosx, sinx,xcosx,xsinx}.

Q4. N/A

Q5. N/A

Q6. N/A

Q7. N/A

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