Linear Transformation Problems

**lllll** · February 3rd 2008, 04:49 PM

I have been stuck on a problem for awhile now and am in need of some help

Let $T:R^2 \rightarrow R^3$ be defined by $T(a_{1},a_{2}) = (a_{1}-a_{2},a_{1},2a_{1}+a_{2})$ . Let $\beta$ be th standard ordered basis for $R^2$ and $\gamma$ = {(1,1,0),(0,1,1),(2,2,3)}. Compute $[T]^{\gamma}_{\beta}$ .

Based on the answer in the back of the book it's supposed to be:

$[T]^{\gamma}_{\beta} = \begin{pmatrix} -1/3 & -1 \\ 0 & 1 \\ 2/3 & 0 \end{pmatrix}$

I have absolutely no idea how they got that solution, since when I input the values of the standard ordered basis I get $\begin{pmatrix} 1 & -1 \\ 1 & 0 \\ 2 & 1 \end{pmatrix}$ .

**tttcomrader** · February 3rd 2008, 07:04 PM

Now, $\beta = \{ e_{1},e_{2} \}$ , so $T( \beta ) = T(e_{1},e_{2}) = T \begin{pmatrix} 1 \\ 0 \end{pmatrix} , T \begin{pmatrix} 0 \\ 1 \end{pmatrix}$

Now, $T \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} = x \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + y \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} + z \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$ , our job is to find x , y, and z.

Well, guess what? Plug in $- \frac {1}{3}, 0, \frac {2}{3}$ in for x, y, z, and see if they agree.

And that is the first half of your answer.

I'm also learning this stuff now in my class, and the matrix representation of a linear transformation is really a b to understand, I kept searching on the internet to find a good example, here is one if you want to see it.

http://www.geocities.com/marry_2000_...inearTrans.pdf

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