Maps in the plane

Determine matrices of the following linear maps in the plane (\(\mathbb R^2\to \mathbb R^2\)) with respect to the canonical base \(K\).

Variant 1
the axis symmetry w.r.t. to axis of the 1st and the 3rd quadrant.
Resolution
It suffices to determine images under \(f\) of the vectors of the canonical basis \(e_1=(1{,}0)^T\) and \(e_2=(0{,}1)^T\) with respect to the same basis.

\([f(e_1)]_K=f(e_1)=(0{,}1)^T\), and similarly \(f(e_2)=(1{,}0)^T\).

The rquired matrix is composed from these vectors \([f]_{KK}=\Big(f(e_1),f(e_2)\Big) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}\).
Variant 2
counterclockwise rotation by \(90^\circ\) around the origin (the first axis is vertical, the second horizontal).
Result
\([f]_{KK}= \begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}\).
Variant 3
counterclockwise rotation by \(\alpha\) around the origin.
Result
\( \begin{pmatrix} \cos \alpha & -\sin\alpha \\ \sin \alpha & \cos\alpha \\ \end{pmatrix}\).
Variant 4
the projection to the first coordinate \(p_1:(x,y)\to(x,0)\)
Result
\( [p_1]_{KK}= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \).