Originally Posted by

**Glitch** The question:

If $\displaystyle {v_1, v_2}$ are linearly independent in a real vector space V and $\displaystyle v_3 = 2v_1 + v_2$, is there a linear map $\displaystyle T:W -> \mathbb{R}^2$ where $\displaystyle W = span(v_1, v_2)$ such that

$\displaystyle T(v_1) = \[

\left( {\begin{array}{c}

1 \\

2 \\

\end{array} } \right)

\]$, $\displaystyle T(v_2) = \[

\left( {\begin{array}{c}

-3 \\

2 \\

\end{array} } \right)

\]$, $\displaystyle T(v_3) = \[

\left( {\begin{array}{c}

-1 \\

3 \\

\end{array} } \right)

\]$?

I'm not sure how to do this. I'm thinking it has something to do with the fact that a linear map needs to satisfy:

$\displaystyle T(\lambda v_1 + ... + \lambda_n v_n) = \lambda_1 T(v_1) + ... + \lambda_n T(v_n)$

Any assistance would be great.