I have given the following three vectors:
$v_1:= \begin{pmatrix}1\\-1\\0\end{pmatrix}, v_2:= \begin{pmatrix}1\\1\\1\end{pmatrix}, v_3:= \begin{pmatrix}0\\1\\1\end{pmatrix} \in \mathbb{C^3}.$
I need to prove, that there is only one linar map defined as:
$\varphi : \mathbb{C^3}\rightarrow \mathbb{C^{2×2}}$ with $ \varphi( v_1 ) = \left( \begin{array}{cc} i & 0 \\ 0 & 0 \end{array} \right), \varphi( v_2 ) = \left( \begin{array}{cc} 1 & 0 \\ -2 & 0 \end{array} \right), \varphi( v_3 ) = \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right).$
After that I need to determine $\varphi( \begin{pmatrix}a\\b\\c\end{pmatrix} )$ for any $a,b,c \in \mathbb{C}$.
I know the general definiton of a linear map but how do I apply these concepts to this task?
My first thought was to prove that the vectors $v_1, v_2, v_3$ are a base. But how do I continue from this step or is this even the wrong approach?
Best Answer
You are done after that step. There should be a theorem somewhere in your textbook/lecture, stating that every linear map is uniquely determined by images of a basis and that every choice of images yields a well defined linear map.
If you don't have that theorem, proof it yourself.
First, show that once you know the images of a basis under a linear map $\phi$, you know the whole map, so you can construct $\phi(v)$ for all vectors $v$ as soon as you know $\phi(b_i)$ for all base vectors $b_i$.
Next, show that every choice $\phi(b_i) := w_i$ can be extended to a linear map. After doing the first part it should be clear how to define the map from the given images of the basis, so what is left to do is to check that this map is indeed linear; but also that will not be that hard because of how the map gets defined from the images of the basis.