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Let $K$ be a field and $ f : V \rightarrow W$ a $K$-linear map between finite dimensional vector spaces over $K$, $V$ and $W$. Show that there exist ordered bases $\mathfrak{B}_V$ of $V$ and $\mathfrak{B}_W$ of $W$ such that $$M_{f,\mathfrak{B}_V ,\mathfrak{B}_W} = \begin{pmatrix} 1_r & 0 \\ 0 & 0 \end{pmatrix}$$

where $r = dim(im(f))$ is the rank of $f$,

$1_r ∈ K^{r×r}$ is the $r × r$ unit matrix,

and $0$ is the fitting zero matrix.

How do I prove the existence of such bases? Can I just take the canonic bases of $V$ and $W$ and note them as tuples? How do I prove they are ordered?

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    If $\;f\;$ is the zero map it is going to be tough to find that $\;1_r\;$ ... I think some restricting conditions must be put. – 2017-01-12
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    @DonAntonio there is nothing to be found about $f$, it is represented by the matrix, which changes for different dimensions. – 2017-01-12
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    @B, I've no idea what you meant by that. I am saying that if $\;f\;$ is the zero map, which is always a linear map between **any** two linear spaces over the same field, then it is going to represented by the zero matrix *in any basis* and, thus, the claim is false. Some further conditions seem to be needed on $\;f\;$ . – 2017-01-12
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    @DonAntonio if $f$ is the zero map then $r=0$. I would say choose a basis for the kernel of $f$ and extend it to a basis of $V$. Then consider its image under $f$ and extend to a basis of $W$. – 2017-01-12

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