Transforming Linear Transformation with Non-Standard Basis

[bmxicle]

Homework Statement

Suppose the matrix standard matrix for a linear trnaformation T: R^2 --->R^2 is[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Cbegin%7Bequation%7D%5Cbegin%7Bpmatrix%7D2%20%26%20-3%20%5C%5C%200%20%26%201%5Cend%7Bpmatrix%7D%5Cend%7Bequation%7D.gif

Find the matrix T with respect to basis B, i.e. find [T]_B. The basis B contains the vectors b₁=(1,1) and b₂=(1,-1).

Homework Equations

T(u + v) = T(u) + T(v)
cT(u) = T(cu)

The Attempt at a Solution

Well I'm definitely feeling like I don't have a solid understanding of what's going on here, but here's some of what I've tried.

So we want to see how the linear transformation transforms a vector (h,k) in the Basis B. My first thought was to see how e₁ and e₂ are transformed and then write them in terms of the basis vectors

So i found that:
e₁ = 1/2b₁ + 1/2b₂
e₂ = 1/2b₁ - 1/2b₂

From the matrix i used that
T(e1) = 2e₁ = 2(1/2b₁ + 1/2 b₂) = b₁ + b₂
T(e2) = -3e₁ + e₂ = -3(1/2b₁ + 1/2b₂) + (1/2b₁ - 1/2b₂) = -b₁ - 2b₂

Now I'm not sure what to do next. Any hints would be great, or suggestions for an online source to read because the section in my textbook on this hasn't helped me to much.

 

[Undoubtedly0]

Greetings! You're close, but instead of seeing how e₁ and e₂ are transformed, you'll want to see how b₁ and b₂ are transformed. Then write T(b₁) and T(b₂) in terms of b₁ and b₂, just like you did for T(e₁) and T(e₂).

 

[Disinterred]

The idea here is that you have a linear transformation that works on vectors in the standard basis i.e. A = a₁e₁ + a₂e₂. But say you have a bunch of vectors, (say millions) in terms of another basis set, which you would like to transform via the transformation matrix. Well you could either find all the components of these vectors in the standard basis and then apply the original transformation matrix. Or the easier way is to transform the matrix so that it can transform vectors in the new basis correctly.

 

[bmxicle]

ahh think I see how to transform the matrix now.

Let D = the transformation matrix with repect to the basis B.
Let C = the change of basis matrix with columns composed of b1, b2
Let A = the standard transformation matrix.

x = C[x]_B and because the columns of C by definition are linearly independent [x]_B = C^-1x

D[x]_B = [Ax]_B = C^{^-1}Ax = C^{^-1}AC[x]_B

which leads to

D = C^-1AC

This gave me the right answer.