- #1

AimaneSN

- 5

- 1

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.

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.

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