Question

1. Let -1 A= -1 1 -2 3 -2 3 4 2 (a) Find a basis for Col(A). (b) Find a basis for Null(A). 2. Show that 1 -4 1 0 9 BE { 2 -5 5 -7 is a basis for W, where W= 2s - 5t 3r + 8 - 2t r - 4s + 3t -r + 2s ER:r, 8, ER 3. Let A= 3 0 0 93 10 -5 0-2 Find eigenvalues of A and determine a basis for each eigenspace. 4. Find a 3 x 3 matrix A having the following three eigenpairs: (8) (8) (8)

Please help with these 4 from my linear algebra study guide, thank you!

Answer 1

0 Given matrix es. - - 1 -2 A = -2 2 - we first reduce the matrix in its row echelon form Applying the row operation. Rq R1+R2 R3 R3-2R, 2 - 1 0 0 - 3 8

RI R, +R2 R3= R₂ + R2 - 0 3 R₂ R2 and R3 3 - O O - 3 RI R, + R3 2 Doo -1 rref (A) a there In rret of A are three leading a I marked in o), corresponding columns in A form (4) اما C01 (A) first three columns of A col (A) = {E}] [] [] 16 since there es no row in rref (A)

so that have all entries zero, the dimension of hull(A) and its basis is an empty set basis of Null (A) = 0