# Innner products and basis representation

1. Aug 18, 2009

### iontail

hi, I have a quickon vector spaces.

Say for example we have

X = a1U1 + a2U2 ....anUn
this can be written as

X = sum of ( i=0 to n) ai Ui

now how can I get and expression of ai in therms of X and Ui.

do we use inner product to do this...ans someone please explain how to go forward.

2. Aug 18, 2009

### HallsofIvy

If the Ui basis is "orthonormal" then, taking the inner product of X with Uk gives $<X, U_n]>= a_1 <U_1,U_k>+ \cdot\cdot\cdot+ a_k<U_k,U_k>+ \cdot\cdot\cdot+ a_n<U_n, U_k>= a_1(0)+ \cdot\cdot\cdot+ a_k(1)+ \cdot\cdot\cdot+ a_n(0)= a_k$.

That is, for an orthonormal basis, $a_k= <X, U_k>$. If the basis is NOT orthonormal, there is no simple formula. That's why orthonormal bases are so popular!

3. Aug 18, 2009

### iontail

the basis is orthonormal...so the solution you suggested should be ok...however i dont have latex and have never used it before so cant view your reply. do I just downlad latex to view the thread or do I have to do something else.

4. Aug 18, 2009

### iontail

5. Aug 18, 2009

### tiny-tim

LaTeX

Hi iontail!

You don't need to "have" LaTeX, it should be visible anyway.

There's just something wrong with that particular LaTeX …I can't read it either

(I can't see what's wrong with the code though.)

To see the original code, just click on the REPLY button.

6. Aug 19, 2009

### Дьявол

Here is what HallsofIvy want to write:

$$<X, U_n]>= a_1 <U_1,U_k>+ \cdot\cdot\cdot+ a_k<U_k,U_k>+ \cdot\cdot\cdot+ a_n<U_n, U_k>= a_1(0)+ \cdot\cdot\cdot+ a_k(1)+ \cdot\cdot\cdot+ a_n(0)= a_k$$