# Change of Basis matrix

## Homework Statement

B1 = {[1,2], [2,1]} is a basis for R2

B2 = {[1,-1], [3,2]} is a basis for R2

Find the change of basis matrix from B1 to B2

[B2 | B1]

## The Attempt at a Solution

For some reason I can not solve this. I keep ending up with the matrix equaling

[-4/5 1/5
3/5 3/5]

Unfortunately this does not work.

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vela
Staff Emeritus
Homework Helper
That looks correct to me. Why do you think it doesn't work?

That looks correct to me. Why do you think it doesn't work?
I know it looks correct, but when I multiply P by B1 I don't get the identity matrix.

vela
Staff Emeritus
Homework Helper
That's because you shouldn't! That matrix takes the representation of a vector relative to B1 and gives you its representation relative to B2.

For example, take the vector [3,3]. In the B1 basis, its representation would be [1,1]1 since

[1,1]1 = (1)[1,2] + (1)[2,1] = [3,3].

In the B2 basis, its representation would be [-3/5, 6/5]2 since

[-3/5, 6/5]2 = (-3/5)[1,-1] + (6/5)[3,2] = [-3/5+18/5, 3/5+12/5] = [3,3]

If you multiply matrix P by [1,1], you'll find you get [-3/5, 6/5]. It converts the B1 coordinates into B2 coordinates.

So think about what multiplying P by [1,2] (the first vector in B1) represents. You should see there's absolutely no reason to think the answer should be [1,0]. Likewise, for the second vector [2,1], you wouldn't expect [0,1].