BMTE-141 English Medium Solved Assignment 2026

Description

BMTE-141 English Medium Solved Assignment 2026 Available

Part-A

Find the angle between the vectors sqrt(2) * i + 2j + 2k and i + sqrt(2) * j + sqrt(2) * k

ii) Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1.1.0).

iii) Check whether W=\ (x,y,z) in R^ 3 | x + y – z = 0 \ is a subspace of R ^ 3

iv) Check whether the set of vectors (1 + x, x + x ^ 2, 1 + x ^ 3) is a linearly independent set of vectors in P_{y} the vector space of polynomials of degree <= 3 . (2)

v) Check whether T / (R ^ 2) -> R ^ 2 defined by T(x, y) = (- y, x) is a linear transformation.

vi) If (v_{1}, v_{2}) is an ordered basis of R ^ 2 and [f_{1}(v), f_{2}(v)] is the corresponding dual basis find f_{1}(2v_{1} + v_{2}) and f_{2}(v_{1} – 2v_{2}) (2)

vii) Find the kernel of the linear transformation T / (R ^ 2) -> R ^ 2 defined by T(x.y)= (2x + 3y * 0.2x – 3y)

viii) Describe the linear transformation T / (R ^ 2) -> R ^ 2 such that where B is the standard basis of R ^ 2 [T] B = [[1, 2], [2, 0]]

ix) Find the matrix of the linear transformation T / (R ^ 2) -> R ^ 2 defined by T(x, y) = (2y, x – y) with respect to the ordered basis \{(0, – 1), (- 1, 0)\} (2)

x) Let A be 12 * 3 matrix, B be 23 * 4 matrix and C’ be a 3 * 2 matrix and D be a matrix. Is AB + C ^ t * D defined? Justify your answer. 3 * 41 (2)

xi) Verify Cayley-Hamilton theorem for the matrix A = [[1, – 1], [0, 2]]

xii) Check whether [[1], [2], [1]] is an eigenvector for the matrix What is the corresponding eigenvalue? [[1, 0, 0, 1], [0, 1, 1, 0], [0, 0, 1, 1], [0, 1, 0, 1]]

xiii) Let C(0,1) be the inner product space of contirous real valued functions on the interval [0, 1] with the inner product Find the inner product of the functions f(t) = 2t g(t) = 1/(t ^ 2 + 5) (2) \langlef, g\rangle = integrate f(t) * g(t) dt from 0 to 1

xiv) Find adjoint of the linear operator T / (C ^ 2) -> C ^ 2 defined by T(z_{1}, z_{2}) = (z_{2}, z_{1} + i*z_{2}) with respect to the standard inner product on c ^ 2 (2) Gi

xv) Find the signature of the quadratic form x_{1} ^ 2 – 2x_{2} ^ 2 + 3x_{3} ^ 2

Part-B (40 Marks)

1) a) Let S be any non-empty set and let V(S) 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) ,+.) a vector space.

Check that B = \{1, 2x + 1, (x – 1) ^ 2\} is a basis for P_{2} the vector space of polynomials with b) (3) real coemciems of degree <= 2

2) a) Let AT / (R ^ 3) R^ 3 | a linear operator and suppose the matrix of the operator with respect to the ordered basis B = \{[[1], [0], [0]] * [[1], [0], [1]] * [[0], [1], [0]]\}

Find the matrix of the linear transformation with respect to the basis in * [[1, 0, – 1], [0, 1, 1], [1, 0, 1]]

B’ = \{[[1], [0], [1]] * [[1], [0], [- 1]] * [[0], [1], [0]]\}

b) Show that V=\ (x, 4x, 3x) in R^ 2 |x in R\ ; a subspace of R ^ 3 Also find a basis for subspace U R ^ 3 J = R ^ 3

3) a) Find the eigenvalues and eigenvectors of the matrix diagonalisable? Justify your answer. Is the matrix B = [[1, 1, 0], [- 1, 3, 0], [1, – 1, 1]]

b) Find Adj(A) where Hence find A = [[3, 2, 2], [- 1, 1, 0], [3, 0, 1]] A ^ – 1

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

2x – y + 2z = 1 x + 2y + z = 3 3x + y + z = 0

b) Find the minimal polynomial of the matrix [[2, 1, 0, 1], [- 1, 0, 0, 1], [- 2, – 2, – 1, 3], [0, 0, 0, 1]]