innner products and basis representation

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.

HallsofIvy

If the U_i basis is "orthonormal" then, taking the inner product of X with U_k gives [itex]<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[/itex].

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

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.

iontail

thanks for the reply...as well.

tiny-tim

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.

Дьявол

Here is what HallsofIvy want to write:

[tex]<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[/tex]