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## Homework Statement

*For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x*

_{1}+ x_{2}, 2x_{1}- x_{2}), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.## Homework Equations

T(v) is given, (x

_{1}+x

_{2}, 2x

_{1}-x

_{2})

## The Attempt at a Solution

Okay, I see that T(v) is simply (2+1, 2*2-1) --> (3, 3), but this matrix business has me a bit confused. From my textbook:

*Using the basis B = {(1, 2), (-1, 1)}, you find that v = (2, 1) = 1(1, 2) - 1(-1, 1), which implies [v]*

_{B}= [1 -1]^{T}.Now, where did this seemingly 'magical' 1 and -1 come from? The matrix relative to B and B' is obviously {(3, 0), (0, 3)}, but I have no idea where this [1 -1]

^{T}came from, nor what it is. I see that this is the coordinate matrix for (3, 3) relative to B, because I see that A*v in this case gives (3, 3), but have no idea how it was derived. Perhaps there's some small gem or concept here that I'm missing?

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