Finding the orthogonal projection of a vector without an orthogonal basis

[AimaneSN]

Hi there,

I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove :

Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E##

Then we have:

##\forall y \in F## : ##y=p_F(x) \Leftrightarrow \forall i= 1,...,p : (x-y,e_i) = 0##

where ##(.,.)## denotes an inner product
and the linear map ##p_F## is the orthogonal projection onto ##F##.

I managed to prove the equivalence only when the family of the vectors ##(e_i)## is orthogonal but the result is more general.

Thank you and I would appreciate any hint or help.

 

[Infrared]

The inner product ##(x-y,e_i)## is zero for all ##i## if and only if ##x-y\in F^\perp## if and only if ##x## is of the form ##x=y+z## where ##z\in F^{\perp}## if and only if ##y## is the orthogonal projection of ##x## onto ##F##.

Your post was a bit of effort to read- it would be better to use latex.

 

[AimaneSN]

Thank you for your reply, it's much clearer now. I just modified my post using Latex code.

 

[WWGD]

OP: Notice you can always use Gram-Schmidt to orthogonalize your basis vectors, and this won't affect the span .