Change of Basis Matrix for R2: B1 to B2

[trojansc82]

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

Homework Equations

[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.

 

[vela]

That looks correct to me. Why do you think it doesn't work?

 

[trojansc82]

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]

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].