Question

Exercise 3. Let u2= (5) C) V2 = V1 = and E u1, u2},F = {v1,v2} be two ordered bases for R2. Let also 5 (i) Find the coordinate vectors of [x]E and [x\f. (ii) Find the transition matrix S from the basis E to F. (ii) Verify that [x]f = S[r]E

Answer 1

3. (i). Let A =

2	1	4
1	2	5

The RREF of A is

1	0	1
0	1	2

This implies that [x]_E = (1,2)^T.

Also, let B =

4	1	4
3	1	5

The RREF of A is

1	0	-1
0	1	8

This implies that [x]_F = (-1,8)^T.

(ii). Let M =

4	1	2	1
3	1	1	2

The RREF of M is

1	0	1	-1
0	1	-2	5

Hence the transition matrix S from the basis E to the basis F is

1	-1
-2	5

(iii). S[x]_E = (-1,8)^T= [x]_F.