<s>[INST] Existence of linear mapping I am studying for an exam in linear algebra and I am having trouble solving the following:
Do linear mappings $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with the following properties exist?
$1)$  $\phi_1 \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} $,   $\phi_1 \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix} $, $\phi_1 \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 \\3 \end{pmatrix} $
$2)$  $\phi_2 \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 0 \\0\end{pmatrix} $,   $\phi_2 \begin{pmatrix} 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 2\\ 3 \end{pmatrix} $
$3)$  $\phi_3 \begin{pmatrix} 1 \\ -1 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} $,   $\phi_3 \begin{pmatrix} -2 \\ 5 \end{pmatrix} = \begin{pmatrix} -3 \\ 1 \end{pmatrix} $, $\phi_3 \begin{pmatrix} 2 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\1 \end{pmatrix} $
I know that the following properties have to hold for a linear mapping:


*

*$f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y})$

*$f(\alpha \mathbf{x}) = \alpha f(\mathbf{x})$

*$f(0) = 0 $


I conclude that $2)$ is a not a linear mapping since $\phi(0)$ is not mapped to $0$.
But how shall I proceed with the others?
 [/INST] 1) You are given the image of $3$ vectors whereas you only need $2$ to define the mapping on $\Bbb R^2$ (because $\Bbb R^2$ is of dimension $2$). So you can assume you defined $\phi$ with the first two and then check if the $3^\text{rd}$ one is true. Now $\begin{pmatrix}1\\2\end{pmatrix}=2\begin{pmatrix}1\\1\end{pmatrix}-\cfrac{1}{2}\begin{pmatrix}2\\0\end{pmatrix}$ so we should have $\begin{pmatrix}0\\1\end{pmatrix}=2\begin{pmatrix}5\\2\end{pmatrix}-\cfrac{1}{2}\begin{pmatrix}2\\3\end{pmatrix}$ which is not the case so the answer is No.
2) As you said, $0$ should be mapped to $0$, and it is not so No.
3) It's the same thing as 1), you need the image of two vectors foming a basis to define your mapping but you have $3$ so you have to check whether the $3^{\text{rd}}$ one is consistent with the two others.
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