Inner Product and Linear Transformation

  • Thread starter J-Wang
  • Start date
  • #1
4
0

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.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
956
Since V is a finite dimensional vector space, there exist an orthonormal basis [itex]\{v_i\}[/itex] for i from 1 to n. Let [itex]a_i= L(v_i)[/itex]. What can you say about <x, v> with [itex]v= \sum_{i=1}^n a_iv_i[/itex]?
 
  • #3
4
0
Does it has something to do with the Projection of x on the basis? :D
 
  • #4
hunt_mat
Homework Helper
1,741
25
Are you trying to prove the Reisz-representation theorem?

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

Related Threads on Inner Product and Linear Transformation

Replies
3
Views
5K
Replies
2
Views
742
  • Last Post
Replies
3
Views
1K
Replies
2
Views
3K
Replies
3
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
1K
Replies
2
Views
2K
Replies
8
Views
7K
Top