<s>[INST] What is the interpretation of the difference? We have the linear maps \begin{equation*}f:\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix} \mapsto \begin{pmatrix}x_1+x_2 \\ x_3 \\ x_1+x_2\end{pmatrix} \ \text{ and } \ h:\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix} \mapsto \begin{pmatrix}x_1+ x_2 \\ x_3 \\ x_1-x_2\end{pmatrix}\end{equation*} 
We also have the vectors \begin{equation*}\vec{v}:=\begin{pmatrix}1 \\ -1 \\ 0\end{pmatrix} \ \text{ und } \ \vec{w}:=\begin{pmatrix}0 \\ 1 \\ 1\end{pmatrix}\end{equation*} 
I want to check if $f(\vec{v}$ and $f(\vec{w})$ are linearly (in)dependent and then the same for $h(\vec{v})$ and $h(\vec{w})$ and then I want to interpret the difference. 
With $f$ we get \begin{equation*}f(\vec{v})=f\begin{pmatrix}1 \\ -1 \\ 0\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 0\end{pmatrix} \ \ \text{ and } \ \ f(\vec{w})=f\begin{pmatrix}0 \\ 1 \\ 1\end{pmatrix}=\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix}\end{equation*} 
The vectors $f(\vec{v})$ and $f(\vec{w})$ are linearly dependent, since $f(\vec{v})=0 \cdot f(\vec{w})$. 
With $h$ we get \begin{equation*}h(\vec{v})=h\begin{pmatrix}1 \\ -1 \\ 0\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ 2\end{pmatrix} \ \ \text{ and } \ \ h(\vec{w})=h\begin{pmatrix}0 \\ 1 \\ 1\end{pmatrix}=\begin{pmatrix}1 \\ 1 \\ -1\end{pmatrix}\end{equation*} 
The vectors $h(\vec{v})$ and $h(\vec{w})$ are linearly independent, since the one vector has zeroes and the other one not, correct? 
What is the interpretation of the difference? 
 [/INST] Yes it is correct, the interpretation is that $v$ belongs to the nullspace of $f$ while by $h$ vectors $v$ and $w$ are mapped into two linearly independent vectors.
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