Assignment -

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

i) The relation ∼ defined by R by x ∼ y if x ≥ y is an equivalence relation.

ii) If S₁ and S₂ are finite non-empty subsets of a vector space V such that [S₁] = [S₂], then S₁ and S₂ have the same number of elements.

iii) For any square matrix A, ρ(A) = det(A)

iv) The determinant of any unitary matrix is 1.

v) If the characteristic polynomials of two matrices are equal, their minimal polynomials are also equal.

vi) If the determinant of a matrix is 0, the matrix is not diagonalizable.

vii) Any set of mutually orthogonal vectors is linearly independent.

viii) Any two real quadratic forms of the same rank are equivalent over R.

ix) There is no system of linear equations over R that has exactly two solutions.

x) If a square matrix A satisfies the equation A² = A, then 0 and 1 are the eigenvalues of A.

Q2) a) Which of the following are binary operations on S = {x ∈ R|x > 0}? Justify your answer.

i) The operation ∇ define by x ∇ y = x(y-2).

ii) The operation Δ defined by x Δ y = e^x+y.

Also, for those operations which are binary operation, check whether they are associative and commutative.

b) Find the equation of the line passing through the point A(1, 0, -1) and parallel to the line joining B(1, 2, 3) and C (-1, 2, 0). Is it perpendicular to the line (1 + 3α, 0, 1 - 2α).

c) Find the radius and the center of the circular section of the sphere |r| = 17 cut off by the plane r·(i + 2j + 2k) = 24.

Q3) a) Let V = R². Define addition + on V by (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂) and scalar multiplication · by r · (a, b) = (ra, 0). Check whether V satisfies all the condition for it to be a vector space over R with respect to these operations.

b) Let V be the vector space of 2 x 2 matrices over R. Check whether the subsets

are subspaces over R. For those sets which are subspaces, find their dimension and a basis over R.

Q4) a) Let T : R⁴ → R⁴ be defined by

T (x₁, x₂, x₃, x₄) = (-x₂, x₁, -x₄, x₃)

Check that T is a linear operator and T⁴ = I. Is T invertible?

b) Let V be a vector space over a field F and let T : V → V be a linear operator. Show T(W) ⊂ W for any subspace W of V if and only if there is a λ ∈ F such that Tv = λv for all v ∈ V.

Q5) a) Check whether the following system of equations has a solution.

x + y + 3z + w = 5

-x + y + z - 5w = 7

X + 2y + 5z - w = 5

b) Let T : P₂ → P₁ be defined by

T (a + bx + cx²) = b + c + (a - c)x.

Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B₁ = {x², x² + x, x² + x + 1] and B₂ = {1, x}. Find the kernel of T.

Q6) a) Check whether the matrices A and B are which are diagonalisable. Diagonalise those matrices which are diagonalizable.

b) Find the minimal polynomials of A and B in part a).

c) Find inverse of matrix B in part A) of the question by finding the adjoint as well as using Cayley-Hamiltion theorem.

Q7) a) Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of R4 generated by the vectors

{(-1, 1, 0, 1), (1, 0, -1, 0), (1, 0, 2, -1)}

b) Consider the linear operator T : C⁴ → C⁴, defined by

T(z₁, z₂, z₃, z₄) = (-iz₂, iz₁, -iz₄, z₃).

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

ii) Check whether T is unitary.

8) a) Find the orthogonal canonical reduction of the quadratic form -x² + y² + z² + 2xy - 2xz + 2yz. Also, find its principal axes.

b) Reduce the conic x² - 6xy + y² - 4 = 0 to standard form. Hence the given conic.