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Working through a homework problem from my linear algebra course. We're using the Gareth Williams Linear Algebra with Application - 7th Edition. The question comes from section 5.2: Matrix Representation of Linear Transformations

Consider the linear transformation $T:R^2 \rightarrow R^2$ defined by $T(x, y)=(2x, x+y)$. Find the matrix $T$ with respect to the basis $\{u_1, u_2\}$ of $R^2$ where: $u_1=(1, 2)$ and $u_2=(0, -1)$

Here is my process thus far, I may be over thinking things a bit.

First I evaluate the transformation on the standard basis $\{e_1, e_2\}$. $T(e_1)=(2, 1)$ and $T(e_2)=(0, 1)$

Then: $c_1u_1 + c_2u_2 = (2, 1)$ and $c_1u_1 + c_2u_2 = (0, 1)$

Solving for $c_1$ and $c_2$ I get the coordinate vectors $[T(u_1)]_B = (2, 3)$ and $[T(u_2)]_B = (0, -1)$

I should then get the transition matrix $T=[[T(u_1)]_B, [T(u_2)]_B]$ (I'm not sure how create matrices but it should be the $2 \times 2$ matrix with column vectors $(2,3)$ and $(0, -1)$)

I'm then to find the image of $(-1, 3)$ which I believe I would just left multiply by $T$. My issue is that both my $T$ and my image are nowhere close to what the solution in the back of the text says they should be. I've led myself astray somewhere here and I'm lost as to where. Any help is appreciated!

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Denote the standard basis by $E=\{e_1=(1,0)^T, e_2=(0,1)^T\}$ and the new basis by $\{u_1=(1,2)^T, u_2=(0,-1)^T\}$ (here I adopt a column vector convention; if you adopt the row vector convention, just take the transpose of everything). There are two ways to tackle the problem.

  1. Compute $T(u_1)$ and $T(u_2)$ first. Then express them as linear combinations of $u_1$ and $u_2$. That is, solve $T(u_1) = au_1+bu_2$ and $T(u_2)=cu_1+du_2$ for the numbers $a,b,c$ and $d$. Then the transformation matrix of $T$ under the new basis is $$ \begin{pmatrix}a&b\\ c&d\end{pmatrix}. $$
  2. Let $S=(u1, u2)$ (i.e put the two columns $u_1, u_2$ together to form a 2x2 matrix). Compute the change-of-basis matrix $S^{-1}$. Then the transformation matrix of $T$ under the new basis is given by $S^{-1}\left(T(e_1), T(e_2)\right)S$.

I am not sure what do the $B$ and the term "transition matrix" refer to, but clearly your $[[T(u_1)]_B, [T(u_2)]_B]=\begin{pmatrix}2&0\\ 3&-1\end{pmatrix}$ is equal to my $\left(T(e_1), T(e_2)\right)S$, whose columns are the results of $T(u_1)$ and $T(u_2)$ under the old basis. So, your answer is missing a left multiplication of $S$.

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You got it - the subscript B is how the text denotes the standard basis –  Ben Anderson Dec 10 '12 at 20:47
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