Well this question has been driving me crazy.
Let B = {[v1,v2]} and C = {[u1,u2]}.
= , =
= , =
Then B and C are bases of the same subspace of .
a) Find the change of basis matrix from B to C.
b) If the coordinate vector x relative to B is = [1 3 , what is the coordinate vector x relative to C?
I dont understand if B and C can be in a subspace of with only two vectors.
Do I have to extend them to 3 vectors? Can I just extend it with a standard vector or do I just leave them as they are.
Well I tried to extend B and C by
B =
C =
Then using the augmented matrix [C B] I used gaussian elimination to get the change of basis matrix D.
D =
I also checked this by working out the inverse of C, so the inverse of C times B = D and I got the same answer to D.
Then I got stuck on question b) because as I understand to work out the coordinate vector x relative to C:
[x = D[x
but I cant multiply D (3x3 matrix) by a 2x1 matrix, so Im not sure do I have to extend the coordinate vectors aswell?
I also attempted it another way by trying to find D = [[v1 , [v2
Where:
2 -1 1 1
1 1 0 1
2 -1 1 1
-----------
-3 1 -1
-3 1 -1
Then I got -3[Row 2] = [1 -1] and back subtituted into the first piviot row and the result was a 3x2 matrix where I could multiply by the coordinate vector relative to B inorder to work out b).
I dont know where I stand as to how to completely answer this question, your help would be much appreciated. I think I may have missed some important step. or thereom. Help.