Linear mapping/Matrix for basis/Surjective and column generating system/Exercise/Solution

We consider the commutative diagramm

$\begin{array}{cccc}{K}^n& \stackrel{{\mathrm{\Psi }}_{𝔳}}{⟶}& V& \\ M_{𝔴}^{𝔳}(\phi )\downarrow & & \downarrow \phi & \\ K^m& \stackrel{{\mathrm{\Psi }}_{𝔴}}{⟶}& W.& \end{array}$

Since the coordinate mappings are bijective, the map $\phi$ is surjective if and only if $M=M_{𝔴}^{𝔳}(\phi )$ is surjective. The image vector of the $j$th standard vector $e_j$ under $M$ is the $j$th column of $M$ and the image space for $M$ is the space generated by the columnns. Therefore, surjectivity is equivalent to the property that the columns form a generating system of $K^m$.

To the exercise