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

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!

 

[iontail]

the basis is orthonormal...so the solution you suggested should be ok...however i don't have latex and have never used it before so can't 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]

LaTeX

...however i don't have latex and have never used it before so can't view your reply. do I just downlad latex to view the thread or do I have to do something else.

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:

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