BMTE-141 Solved Assignment January 2026 | Linear Algebra | IGNOU BSCG

Q: Answer the following questions from Part C.

Answer available — download the full PDF for complete details.

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

Answer available — download the full PDF for complete details.

Q1. Answer the following questions from Part A.

(i)) Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k. (100 words)
(ii)) Find the vector equation of the plane determined by the points (1, 0, −1), (0, 1, 1) and (-1,1,0). (100 words)
(iii)) Check whether W = {(x, y, z) ∈ R³ |x + y − z = 0 } is a subspace of R3. (100 words)
(iv)) 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. (100 words)
(v)) Check whether T: R2 → R2, defined by T(x, y) = (−y, x) is a linear transformation. (100 words)
(vi)) If {U1, U2} is an ordered basis of R2 and { f₁ (v), f2 (v)} is the corresponding dual basis find f1 (201 + 2) and f2 (U1 – 202). (100 words)
(vii)) Find the kernel of the linear transformation T: R2 → R2 defined by T(x, y) = (2x + 3y, 2x – 3y). (100 words)
(viii)) Describe the linear transformation T: R2 → R2 such that [T]B = [[1, 2], [2, 0]] where B is the standard basis of R2. (100 words)
(ix)) 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)}. (100 words)
(x)) Let A be a 2 × 3 matrix, B be a 3 x 4 matrix and C be a 3 × 2 matrix and D be a 3 × 4 matrix. Is AB + CD defined? Justify your answer. (100 words)
(xi)) Verify Cayley-Hamilton theorem for the matrix A = [[1, -1], [0, 2]]. (100 words)
(xii)) Check whether [[1], [1], [1], [1]] is an eigenvector for the matrix [[1, 0, 0, 1], [0, 1, 1, 0], [0, 0, 1, 1], [0, 1, 0, 1]]. What is the corresponding eigenvalue? (100 words)
(xiii)) Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product <f,g> = ∫₀¹ f(t)g(t) dt. Find the inner product of the functions f(t) = 2t, g(t) = 1/(t² + 5). (100 words)
(xiv)) 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. (100 words)
(xv)) Find the signature of the quadratic form x² - 2x2 + 3x3 (100 words)

Key Points:

Angle between vectors: `cos θ = (u ⋅ v) / (||u|| ||v||)` for u and v.
Subspace conditions: Contains zero vector, closed under addition, and scalar multiplication.
Linear Independence: Only trivial solution (all zeros) for `c₁v₁ + ... + cₙvₙ = 0`.
Linear Transformation: Satisfies `T(u+v) = T(u)+T(v)` and `T(cu) = cT(u)`.

Answer: This document provides comprehensive answers to the 15 sub-questions from Part A of the BMTE-141 Linear Algebra course, covering fundamental concepts such as vector operations, plane equations, subspace verification, linear independence, linear transformations, dual bases, kernel computation, matrix representation of transformations, matrix algebra, Cayley-Hamilton theorem, eigenvectors, inner products, adjoint operators, and quadratic form signatures. Each solution includes step-by-step derivat...

Q2. Answer the following questions from Part B.

(a)) 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 (a. f)(x) = af (x). Check that (V(S), +, ·) is a vector space. (350 words)
(b)) Check that B = {1,2x + 1, (x – 1)2} is a basis for P2, the vector space of polynomials with real coefficients of degree ≤ 2. (150 words)

Key Points:

A vector space must satisfy ten specific axioms related to addition and scalar multiplication.
The set of real-valued functions V(S) forms a vector space under pointwise addition and scalar multiplication.
The zero function, 0(x)=0, acts as the additive identity in V(S).
P2 represents the vector space of polynomials of degree at most 2.

Answer: The following sections comprehensively address the given sub-questions from Part B of the BMTE-141 Linear Algebra course, focusing on the definition of a vector space and the concept of a basis, drawing upon fundamental principles typically covered in the module. Vector spaces are foundational structures in linear algebra, providing a framework for operations involving vectors and scalars. Understanding how to verify if a set with defined operations constitutes a vector space is crucial. Simila...

Q3. Answer the following questions from Part B.

(a)) Let T: R3 → R³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis [[1, 0, 0], [0, 1, 0], [0, 0, 1]] is [[1, 1, 0], [1, 0, 0], [0, 1, 0]]. Find the matrix of the linear transformation with respect to the basis B' = [[1, 1, 0], [0, 0, 1], [1, 0, 1]]. (300 words)
(b)) Show that W = {(x, 4x, 3x) ∈ R2|x ∈ R} is a subspace of R³. Also find a basis for subspace U of R³ which satisfies W ⊕ U = R3. (200 words)

Key Points:

Change of Basis Formula: B = P⁻¹AP transforms a matrix A from standard basis to basis B'.
Change of Basis Matrix P: Columns are the new basis vectors expressed in the old basis.
Subspace Criteria: A non-empty set is a subspace if closed under addition and scalar multiplication.
Direct Sum (W ⊕ U = R³): Implies W ∩ U = {0} and Dim(W) + Dim(U) = Dim(R³).

Answer: This response addresses two fundamental concepts in Linear Algebra: changing the basis of a linear operator's matrix representation and identifying subspaces along with their complementary direct sums. Understanding these concepts is crucial for analyzing linear transformations in different coordinate systems and decomposing vector spaces into simpler components, as detailed in BMTE-141. Part (a) involves applying the change of basis formula for a linear operator, which requires careful calcula...

Q4. Answer the following questions from Part B.

(a)) Find the eigenvalues and eigenvectors of the matrix B = [[1, 1, 0], [-1, 3, 0], [1, -1, 1]]. Is the matrix diagonalisable? Justify your answer. (250 words)
(b)) Find Adj(A) where A = [[3, 2, 2], [-1, 1, 0], [3, 0, 1]]. Hence find A−1. (250 words)

Key Points:

Eigenvalues are roots of the characteristic equation det(A - λI) = 0.
Eigenvectors 'v' satisfy (A - λI)v = 0 for a given eigenvalue λ.
A matrix is diagonalizable if its algebraic and geometric multiplicities match for all eigenvalues.
Geometric multiplicity of an eigenvalue is the dimension of its eigenspace, always ≤ algebraic multiplicity.

Answer: This question assesses the fundamental concepts of eigenvalues, eigenvectors, matrix diagonalisation, and matrix inversion using the adjoint method, which are core topics in linear algebra. Understanding these concepts is crucial for solving systems of linear equations, transforming bases, and analyzing linear transformations. Sub-question (a) requires calculating eigenvalues and their corresponding eigenvectors, then determining diagonalisability based on the relationship between algebraic and...

Q5. Answer the following questions from Part B.

(a)) Solve the folowing set of simultaneous equations using Cramer's rule: x+2y + z = 3; 2x - y + 2z = 1; 3x + y + z = 0 (250 words)
(b)) Find the minimal polynomial of the matrix [[2, 1, 0, 1], [-1, 0, 0, 1], [-2, -2, -1, 3], [0, 0, 0, 1]]. (250 words)

Key Points:

Cramer's Rule uses determinants (D, Dx, Dy, Dz) to solve systems of linear equations.
A unique solution exists via Cramer's Rule if the determinant of the coefficient matrix (D) is non-zero.
The characteristic polynomial p(λ) of a matrix A is det(A - λI), whose roots are the eigenvalues.
The minimal polynomial m(λ) is the monic polynomial of least degree such that m(A) = 0.

Answer: This response comprehensively addresses the given Linear Algebra problems from BMTE-141, focusing on Cramer's Rule for solving simultaneous equations and finding the minimal polynomial of a matrix. Each sub-question adheres to the specified word limits and includes detailed, step-by-step calculations and explanations, drawing upon core concepts from linear algebra often covered in this course. Proper academic terminology and presentation are maintained throughout. Cramer's Rule is a fundamental...

Q6. Answer the following questions from Part C.

(a)) 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 ao cos x + a₁ sin x + a2x cos x + a3x sin x. (Put x = 0, π, π/2, π/4 etc. and find a₁. )) (200 words)
(b)) Consider the linear operator T: C3 → C³, defined by T (Z1, Z2, Z3) = (Z1 – iz2, iz₁ – 222 + izz, -iZ2 + Z3). i) Compute T* and check whether Tis self-adjoint. ii) Check whether Tis unitary. (300 words)

Key Points:

Linear independence of functions: A set of functions is linearly independent if their linear combination equaling the zero function implies all scalar coefficients are zero.
Method for function independence: Substitute specific x values (e.g., 0, π, π/2) into the linear combination and solve the resulting system of equations for coefficients.
Adjoint operator (T*): For a linear operator T, its matrix representation A* is the conjugate transpose of the matrix A corresponding to T.
Self-adjoint operator: An operator T is self-adjoint if T = T*, meaning its matrix A is equal to its conjugate transpose A* (A = A*).

Answer:

Q7. Answer the following questions from Part C.

(a)) 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) = x² - 4x1x2 + 2x2x3 + x2 + x3 find the representation of Q in terms of (У1, У2, Уз). (150 words)
(b)) Find the orthogonal canonical reduction of the quadratic form −x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes. (350 words)

Key Points:

Quadratic form transformation requires expressing old coordinates in terms of new ones.
The change of basis matrix P relates coordinates X and Y via X = PY.
Orthogonal canonical reduction simplifies a quadratic form using eigenvalues.
The symmetric matrix A represents the quadratic form (Q(X) = XᵀAX).

Answer: This answer addresses the transformation of a quadratic form between different bases and its orthogonal canonical reduction, concepts central to Linear Algebra, particularly in advanced topics like inner product spaces and diagonalization of matrices, as covered in courses like BMTE-141. The first part focuses on changing the coordinate system, while the second part deals with simplifying the quadratic form through orthogonal transformations to its principal axes.

Q8. 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 Vand dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W1 ∩ W2 ≠ {0}. (100 words)
(ii)) If Vis a vector space and S = {U1, U2, ..., Un } C V, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set. (100 words)
(iii)) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T₁。T2 is invertible, T₂。T₁ is also invertible. (100 words)
(iv)) If an n × n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial. (100 words)
(v)) 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. (100 words)

Key Points:

Dim(W₁+W₂) = Dim(W₁)+Dim(W₂)-Dim(W₁∩W₂) is crucial for subspace intersection problems.
Distinct vectors are not automatically linearly independent; a counterexample is often needed.
For finite-dimensional spaces, an operator is invertible iff it's injective (one-to-one) or surjective (onto).
If T₁∘T₂ is invertible, then both T₁ and T₂ must be individually invertible.

Answer:

BMTE-141 Solved Assignment — January 2026 / July 2026

Linear Algebra | IGNOU Bachelor of Science (General) (CBCS) (BSCG)

BMTE-141—January 2026 / July 2026

Linear Algebra

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BMTE-141 Assignment Questions — January 2026

Q1. Answer the following questions from Part A.

Q2. Answer the following questions from Part B.

Q3. Answer the following questions from Part B.

Q4. Answer the following questions from Part B.

Q5. Answer the following questions from Part B.

Q6. Answer the following questions from Part C.

Q7. Answer the following questions from Part C.

Q8. Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

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