Linear Algebra Answers

Suppose that A and B are 4 × 4 matrices such that B is non-singular. Which of the following statement(s) is/are correct?

1. det(−3A ^T ) = 3⁴ det(A) and det(A) = det(BAB⁻¹ ).

2. det(3B ⁻¹ ) = 3⁴ det(B) and det(A) = det(BAB⁻¹ ).

3. det(2A ^T ) = −16 det(A) and det(A) = det(BAB⁻¹ ).

4. det(2A ^T ) = 2³ det(A) and det(A) = det(BAB⁻¹ ).

Let B =(a1, a2, a3) be an ordered basis of 

R3 with a1 = (1, 0, -1), a2 = (1, 1, 1), 

a3 = (1, 0, 0). Write the vector v = (p,q,r)) as 

a linear combination of the basis vectors 

from B

Let V = Mn(C) and U be the subspace of all Hermitian matrices over C. Find a basis and

dimension of U.

Find all possible jordan canonical form for the matrices whose characteristics polynomial p(t) and minimal polynomial m(t) are p(t)=(t-2)⁴(t-3)² m(t)= (t-2)²(t-3)²

Suppose T:V tends to V is a linear operator. Let W be a sub space of a vector space V.Let W be invariant under the linear operator T₁:V tends to V and T₂: V tends to V.Then prove that W is also invariant under T₁+T₂ and T₁T₂.

Find a matrix A whose minimal polynomial is t⁴-5t³-2t²+7t+4

let W be an inner product space and let w1 and w2 be vectors in W. suppose ll w1 ll =5^1/2 and ll w2 ll= 4 and the angle between w1 and w2 is pi/3. compute

• <w1,w1> and <w2,w2>
• <w1, w2>
• <w1+ 2w2, 3w1-w2>

Let {(1, 1, 1, 1), (1, 2, 1, 2)} be linearly independent which is a subset of vector space V₄. Extend it to a basis of V₄