I would like to get a simple demo to proove that a projection defined for example by : $\vec{v}\vec{v^{*}}\vec{w}$ where $\vec{w}$ is projected onto vector $\vec{v}$.
First, if I start from the definition of an inner product between $X$ and $Y$ vectors, the definition is :
$$\Phi(X,Y)=X^T M Y$$
with $M$ the tensor metric (that I assimilate to the projection matrix $\vec{v}\vec{v^{*}}$ above) :
If I introduce a transformation matrix $P$ to do the link between $(e'_{i})$ and $(e'_{i})$ such as for vectors $X$ and $Y$, I can write :
$$X=PX'$$
$$Y=PY'$$
Introduce this in definition of bilinear form $\Phi$ :
$$\Phi(X,Y) = \Phi(PX',PY')= <X'^{T} P^{T} M P Y'>$$
Any help is welcome
EDIT 2 : The formula $P^{T} M P$ seems to be the general expression of matrix $M$ into ($\vec{e'_i}$) vectors basis : is it correct ?