If I have an equation that involves both dot product and a matrix multiplication, then what are the rules for exchanging operations between them? For instance, the equation for projection of a vector b onto a is often written as:
$$xa=\frac{a^Tb}{a \cdot a} $$
Then, the "derivation" of the projection matrix is done by:
$$Pb = xa$$
$$Pb = \frac{a^Tb}{a \cdot a}a = a\frac{a^Tb}{a \cdot a} = \frac{aa^T}{a \cdot a}b$$
$$ \therefore P = \frac{a a^T}{a \cdot a} $$
Is anything allowed so long as we respect the rules of matrix multiplication? In particular, I am looking at the second step, where we move the scalar c around, $ca = ac$; it seems hacky to me.