I need a perfect answer and detailed explanation for this question. It is easier for me to understand and do revision. I will give the best rate for you. Thank you

Course: Linear Algebra

Answer ALL questions.

1. Let 7: M22 -> P2 be a linear transformation such that =1 - x 0 0 TO ) = 1 +x4x2 T ( ) = 2x - x2 T ( )= 2+x-2x2 Find (a ) Tra c d (b) Tra 2a 3a 4a [5 marks] 2. Let 7: R' - Ra be a linear transformation such that T(1,2,1) = (1,0) T(2,9,0) = (-1,1) T(3,3,4) = (0,1) Find (a) T(x, y, z) (b) T(x, 2x, 3x) [5 marks]3. Given matrix a b C A = (0 d e). 0 0 (a) Find the eigenvalue(s) of A. (b) Find the corresponding eigenvector(s) of A. (c) Find the corresponding eigenspace(s) of A. [5 marks] 4. Given matrix 3 -1 -1 B = (-12 0 5 ) . 4 -2 -1 (a) Find the eigenvalue(s) of B. (b) Find the eigenvalue(s) of B2. (c) What conclusion can you make by comparing answers from part (a) and (b)? [5 marks]