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If $M_2(\mathbb{R})$is the space of all $2\times 2$ real matrices then define:

$$F:M_2(\mathbb{R})\rightarrow M_2(\mathbb{R})$$

...by:

$$F(A)=A+A^{T}$$

...where $A^T$ is the transpose.

For this function $F$:

$\dim(Null(F))=0$ and $\dim(F(M_2(\mathbb{R})))=4$

$\dim(Null(F))=1$ and $\dim(F(M_2(\mathbb{R})))=3$

$\dim(Null(F))=1$ and $\dim(F(M_2(\mathbb{R})))=2$

$F$ is not even a linear transformation