Inner Product and Linear Transformation

[J-Wang]

Homework Statement

Let V be a finite-dimensional real inner product space with inner product < , >.

Let L:V->R be a linear map.

Show that there exists a vector u in V such that L(x) = <x,u> for all x in V.


2. The attempt at a solution
It seems really simple but I just can't phrase it properly.

I know since L is linear, I can express as L(x+y) = L(x) + L(y)

I tried to make x and u be expressed as combination of orthogonal basis, and try to work in the fact that inner product will result in ZERO since they are orthogonal.

Any help would be greatly appreciated.

 

[HallsofIvy]

Since V is a finite dimensional vector space, there exist an orthonormal basis \{v_i\} for i from 1 to n. Let a_i= L(v_i). What can you say about <x, v> with v= \sum_{i=1}^n a_iv_i?

 

[J-Wang]

Does it has something to do with the Projection of x on the basis? :D

 

[hunt_mat]

Are you trying to prove the Reisz-representation theorem?

I would see how this linear functional acted on individual basis elements.