Linear transformation explanation

Question

I have the following exercise:

Consider the linear transformation L: ℝ³→ ℝ². Knowing that:

$$ L \begin{pmatrix}1\\1\\0\end{pmatrix} = \begin{pmatrix}1\\2\end{pmatrix} \space\space\space\space\space\space L \begin{pmatrix}0\\1\\1\end{pmatrix} = \begin{pmatrix}1\\1\end{pmatrix} \space\space\space\space\space\space L \begin{pmatrix}1\\1\\1\end{pmatrix} = \begin{pmatrix}1\\3\end{pmatrix} $$ determine the following: $$ L \begin{pmatrix}3\\4\\2\end{pmatrix} = \space\space ? $$

Can someone explain how to solve this?

Answer 1

$$\begin{pmatrix} 3\\4\\2 \end{pmatrix} = 2\begin{pmatrix}1\\1\\0 \end{pmatrix} + 1 \begin{pmatrix}0\\1\\1 \end{pmatrix} + 1\begin{pmatrix}1\\1\\1 \end{pmatrix}$$

So: $$L\begin{pmatrix} 3\\4\\2 \end{pmatrix} = L \left(2\begin{pmatrix}1\\1\\0 \end{pmatrix} + 1 \begin{pmatrix}0\\1\\1 \end{pmatrix} + 1\begin{pmatrix}1\\1\\1 \end{pmatrix} \right) = 2L\begin{pmatrix}1\\1\\0 \end{pmatrix} + L\begin{pmatrix}0\\1\\1 \end{pmatrix} + L\begin{pmatrix}1\\1\\1 \end{pmatrix} = \begin{pmatrix} 4 \\ 8 \end{pmatrix}$$