Question

linear alegbra

(8) Let B = 2-21。10 (a) Show that B is a basis for span(1121|판이,) (b) Let ขึ-11 and find Ple and Fhe Make sure to show work.

Answer 1

8. (a). Let A be the matrix with the vectors in the given set B as columns. Then A =

1	4	0
2	2	0
3	1	1

The RREF of A is I₃ which implies that the vectors in the set B are linearly independent. Therefore, the set B is a basis for span{(1,2,3)^T,(4,2,1)^T,(0,0,1)^T}.

(b). Let M = [A|v] =

1	4	0	2
2	2	0	1
3	1	1	3

The RREF of M is

1	0	0	0
0	1	0	1/2
0	0	1	5/2

This implies that v = 0(1,2,3)^T+(1/2)(4,2,1)^T+(5/2)(0,0,1)^T so that [v]_B = (0,1/2,5/2)^T .

Let N = [ ɛ|B] =

1	0	0	1	4	0
0	1	0	2	2	0
0	0	1	3	1	1

The matrix N is already in its RREf. Hence, the trasition matrix from the basis B to the standard basis ɛ is P_B→ɛ =

1	4	0
2	2	0
3	1	1