Find The Linear Transformation There will be 3 vectors in $R^3$ $V_1= \begin{pmatrix} 1\\ 2\\ 3\\ \end{pmatrix} $ , $V_2= \begin{pmatrix} 1\\ 1\\ 1\\ \end{pmatrix} $, $V_3= \begin{pmatrix} 1\\ 4\\ 7\\ \end{pmatrix}$ 
and 3 vectors in $R^2$ $W_1=\begin{pmatrix} 1\\ 0\\ \end{pmatrix} $ $W_2=\begin{pmatrix} 0\\ 1\\ \end{pmatrix} $  $W_3=\begin{pmatrix} 3\\-2\\ \end{pmatrix} $
find $T$ for $T(V_i)=(W_i)$
I know T needs to be a $R^{2x3}$ but if I use the elementary basis of $R^3$ I get a $R^{3x3}$
 A: If $V$ and $W$ are any two finite dimensional vector spaces and $b_i$ is a basis of $V$ and $T: V \to W$ is a linear transformation then the columns of $T$ are $Tb_1, Tb_2, \dots$ etc. This is because in the basis $b_i$ the vectors $b_i$ have the coordinates $(1,0,0 \dots), (0,1,0,\dots),$ etc. 
Of course, the entries of the matrix of $T$ when you write it down on paper depend on whatever basis (of $W$) the $Tb_i$ are given in.   
Now for the problem at hand: Note that $V_i$ form a basis of $\mathbb R^3$. Also note that you are given $TV_i$ with respect to the standard basis $(1 0), (0 1)$ of $\mathbb R^2$. As a consequence, the matrix of $T$ with respect to the basis $V_i$ and the standard basis $e_1,e_2$ equals $\widetilde{T}=(W_1W_2W_3)$ (the matrix with columns $W_i$).
To obtain the matrix of $T$ with respect to the standard bases of $\mathbb R^3$ and $\mathbb R^2$ all that remains to be done is this: find the matrix $B$ that maps a vector $v$ given with respect to the standard basis of $\mathbb R^3$ to $v$ given with respect to $V_i$ and compute $\widetilde{T} \cdot B$.
But $B$ is simply the matrix $(V_1V_2V_3)$.
