Linear mapping/Kernel and image/Direct sum/Exercise

Let ${}A={\begin{pmatrix}3&1&5\\-1&0&2\end{pmatrix}}$, and let

$f\colon \mathbb {R} ^{3}\longrightarrow \mathbb {R} ^{2},x\longmapsto A\cdot x$

denote the corresponding linear mapping.

1. Determine a basis and the dimension of ${}\operatorname {kern} f$, and of ${}\operatorname {Im} f$.
2. Find a linear subspace ${}V\subseteq \mathbb {R} ^{3}$ such that

   ${}\mathbb {R} ^{3}=V\oplus \operatorname {kern} f\,$

   holds.

3. Does there also exist a linear subspace ${}U\subset \mathbb {R} ^{2}$, ${}U\neq 0$, such that ${}\mathbb {R} ^{2}=U\oplus \operatorname {Im} f$?

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