matrix representing L

**antman** · April 25th 2009, 10:12 AM

Let L: $R^{2}->R^{2}$ be defined by $L(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix }x+2y\\2x-y\end{pmatrix}$ .
Let S be the natural basis for $R^{2}$ and let $T={\begin{pmatrix}-1\\2\end{pmatrix},\begin{pmatrix}2\\0\end{pmatrix} }$ be another basus for $R^{2}$ . Find the matrix representing L with respect to:
a) S
b) S and T
c) T and S
d) T
e) Compute L $(\begin{pmatrix}1\\2\end{pmatrix})$ using the definition of L and also using the matrices obtained in (a), (b), (c) and (d).

Any help would be incredibly appreciated or an example of how a problenm like this is to be done.

**ThePerfectHacker** · April 25th 2009, 10:29 AM

Originally Posted by antman

Let L: $R^{2}->R^{2}$ be defined by $L(\begin{pmatrix}x\\y\end{pmatrix})=\begin{pmatrix }x+2y\\2x-y\end{pmatrix}$ .
Let S be the natural basis for $R^{2}$ and let $T={\begin{pmatrix}-1\\2\end{pmatrix},\begin{pmatrix}2\\0\end{pmatrix} }$ be another basus for $R^{2}$ . Find the matrix representing L with respect to:
a) S
b) S and T
c) T and S
d) T
e) Compute L $(\begin{pmatrix}1\\2\end{pmatrix})$ using the definition of L and also using the matrices obtained in (a), (b), (c) and (d).

Any help would be incredibly appreciated or an example of how a problenm like this is to be done.

To find the matrix to the standard basis compute $L(0,1),L(1,0)$ as a coloumn vector and construct the matrix $[ L(0,1) | L(1,0) ]$ .

To find the matrix with respect to $T$ compute $[ L(-1,2)]_T, [L(2,0)]_T$ where $[ ~ ~ ]_T$ means you compute the coordinate of that coloum vector with respect to $T$ . Then form the matrix of those coloumn vectors as you done so above.

**antman** · May 1st 2009, 01:14 PM

This question is still not making sense to me. I apologize. I'm trying to learn way too much in such a short amount of time. S is the standard basis that equals $\begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}1\ \0\end{pmatrix}$ ? I am really clueless.

**ThePerfectHacker** · May 1st 2009, 01:19 PM

Originally Posted by antman

S is the standard basis that equals $\begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}1\ \0\end{pmatrix}$ ? I am really clueless.

Almost, you just go it backwards.

The standard basis for $\mathbb{R}^2$ is $(1,0),(0,1)$ (in that order). The standard basis for $\mathbb{R}^3$ is (if you taken Calculus 3 you seen this before) $\bold{i},\bold{j},\bold{k}$ where $\bold{i}=(1,0,0), \bold{j}=(0,1,0), \bold{k}=(0,0,1)$ . In general for $\mathbb{R}^n$ define $\bold{e}_j = (0,0,...,1,0,...,0)$ (here $1\leq j\leq n$ ) where $1$ is in the $j$ -th position, then the standard basis is $\bold{e}_1,\bold{e}_2,...,\bold{e}_n$ . It is easy to express everything in terms of a standard basis because $(a_1,...,a_n) = a_1\bold{e}_1 + ... + a_n \bold{e}_n$ .

**antman** · May 1st 2009, 09:38 PM

For the matrix representing L with respect to S, I did $L(v_{1})={(1+2(0)),(2(1)-0)}=(1,2)$ and $L(v_{2})=(2,1)$ to get $\begin{pmatrix}1&2\\2&-1\end{pmatrix}$ .

Using this process for s, to find matrix with respect to T, gave me (3,-4) and (2,4) that i put into matrix $\begin{pmatrix}-1&2&3&2\\2&0&-4&4\end{pmatrix}$ that i reduced and took right side to get matrix with respect to $T=\begin{pmatrix}-2&2\\1/2&2\end{pmatrix}$ .

For S and T, I got the matrix $\begin{pmatrix}3&-4\\2&4\end{pmatrix}$ and $\begin{pmatrix}3&2\\-4&4\end{pmatrix}$ for T and S.

I am more confident in my answers to part a (S) and part d (T) because I really don't know how to do the others. Can someone help point me in the right direction?

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