# Inner Product and Linear Transformation

#### J-Wang

1. The problem statement, all variables and given/known data
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.

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#### 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

Homework Helper
Are you trying to prove the Reisz-representation theorem?

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

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