Find (linear) transformation matrix using the fact that the diagonals of a parallelogram bisect each other. This is the first time I post something on this website. I'm on this question already for hours. I'm clearly not asking the community to do my homework, I'm hoping someone can explain me how I should solve the following question;

Let $l$ be a line through the origin in $\mathbb{R}^2$, $P_l$ the linear transformation that projects a vector onto $l$, and $F_l$ the transformation that reflects a vector in $l$.


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*Draw diagrams to show that $F_l$ is linear. Diagrams? How does this look like? A standard matrix?

*Figure 3.14 (see image) suggests a way to find the matrix of $F_l$, using the fact that the diagonals of a parallelogram bisect each other. Prove that $F_l = 2P_l(x) - x$, and use this result to show that the standard matrix of $F_l$ is (see image).

*If the angle between $l$ and the positive $x$-axis is $A$, show that the matrix of $F_l$ is (see image).

I attached the question as image
Hopefully you can help.
Thanks!
Image: i.stack.imgur.com/vFkmM.jpg
EDIT: Image shown here:

 A: For the first part, the diagram which you should draw is similar to the figure 3.14.  Your map $F_\ell : \mathbb{R}^2 \to \mathbb{R}^2$ is linear if $F_\ell(x+y) = F_\ell(x) + F_\ell(y)$ and $F_\ell(cx) = cF_\ell(x)$.  Here $x$, $y$ are points in your domain $\mathbb{R}^2$ and $c$ is a scalar.  You can check the equalities by drawing the vectors on either side of the two equations I wrote.  For example, to check the second equality, you have $x$, $\ell$, and $F_\ell(x)$ already drawn in Figure 3.14.  Try drawing $cx$, then $F_\ell(cx)$, and draw $cF_\ell(x)$, and verify that these latter two are equal.  You can do something similar for the first equality.
For part b: I'd prefer to rewrite the equality as $x + F_\ell(x) = 2P_\ell(x)$.  Now look at Figure 3.14.  How is the left side, $x + F_\ell(x)$, depicted geometrically?  How is $2P_\ell(x)$ depicted geometrically?  Convince yourself that these vectors are equal.
Now the standard matrix of a linear map from $\mathbb{R}^2$ to $\mathbb{R}^2$ consists of two columns: the first column says what the map does to the vector $\langle 1, 0 \rangle$, and the second column says what the map does to the vector $\langle 0, 1 \rangle$.  You should first write the standard matrix for $P_\ell$ by actually computing the vector projection of each of these two basis vectors onto the vector $\langle d_1, d_2 \rangle$.  Now you can use the equality to compute the standard matrix of $F_\ell$.
Finally, part c will follow from b if you plug in the direction vector $\langle \cos \theta, \sin \theta \rangle$.
