1. ## Linear Transformations

let e1 = $\displaystyle \begin{bmatrix}1 \\ 0 \end{bmatrix}$, e2 =$\displaystyle \begin{bmatrix}0 \\ 1 \end{bmatrix}$, y1= $\displaystyle \begin{bmatrix}2 \\ 5 \end{bmatrix}$,y2 = $\displaystyle \begin{bmatrix}-1 \\ 6 \end{bmatrix}$

and T: R^2 --> R^2 be a linear transformation that maps e1 into y1 and maps e2 into y2.

find the images of $\displaystyle \begin{bmatrix}5 \\ -3 \end{bmatrix}$ and $\displaystyle \begin{bmatrix}x1 \\ x2\end{bmatrix}$

Ive been messing with this for awhile, I simply dont understand what its asking, or what I am supposed to do.

help!

2. You are given that $\displaystyle T\left( {\left[ \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \right]} \right) = \left[ \begin{gathered} 2 \hfill \\ 5 \hfill \\ \end{gathered} \right]\,\& \,T\left( {\left[ \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right]} \right) = \left[ \begin{gathered} - 1 \hfill \\ 6 \hfill \\ \end{gathered} \right]$.
You know that $\displaystyle \left[ \begin{gathered} 5 \hfill \\ - 3 \hfill \\ \end{gathered} \right] = 5\left[ \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \right] - 3\left[ \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right]$
Now apply the linear transformation.

3. Originally Posted by Plato
You are given that $\displaystyle T\left( {\left[ \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \right]} \right) = \left[ \begin{gathered} 2 \hfill \\ 5 \hfill \\ \end{gathered} \right]\,\& \,T\left( {\left[ \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right]} \right) = \left[ \begin{gathered} - 1 \hfill \\ 6 \hfill \\ \end{gathered} \right]$.
You know that $\displaystyle \left[ \begin{gathered} 5 \hfill \\ - 3 \hfill \\ \end{gathered} \right] = 5\left[ \begin{gathered} 1 \hfill \\ 0 \hfill \\ \end{gathered} \right] - 3\left[ \begin{gathered} 0 \hfill \\ 1 \hfill \\ \end{gathered} \right]$
Now apply the linear transformation.
got it!