Linear Algebra Answers

Suppose T € L(V ) and dim range T =k. Prove that T has at most k + 1distinct eigenvalues.

Find vectors u; v € R^2 such that u is a scalar multiple of (1; 3), v is orthogonal to (1; 3); and (1; 2) =u+ v.

Give an example, with justification, of a 

skew-Hermitian operator on C².

 Let P_k = {p(x)|p(x) is a polynomial of 

degree ≤ k with real coefficients}, for k €N.

Apply the Fundamental Theorem of 

Homomorphism to prove that p₅/p₃ is isomorphic to p₁.

Give an example which satisfies the properties of vector space, subspace and inner product space and number should be complex

Let V and W be two vector spaces over the field F and T, T1, T2 be linear transformations from V to W. Prove the following:

(a) rank(αT) = rank(T), for all α ∈ F and α 6= 0.

(b) |rank(T1) − rank(T2)| ≤ rank(T1 + T2) ≤ rank(T1) + rank(T2

 Suppose u; v € V. Prove that

||au + bv|| = ||bu +

av|| for all a; b € R if and only if ||u|| = ||

v||

.

Find the characteristic equation of the matrix A =





2 1 1

0 1 0

1 1 2



 and hence find the matrix

represented by A8 − 5A7 + 7A6 − 3A5 + A4 − 5A3 + 8A2 − 2A + I.

Let V and W be two vector spaces over the field F and T, T1, T2 be linear transformations from

V to W. Prove the following: (2+3=5 marks)

(a) rank(αT) = rank(T), for all α ∈ F and α 6= 0.

(b) |rank(T1) − rank(T2)| ≤ rank(T1 + T2) ≤ rank(T1) + rank(T2).