Innner products and basis representation

iontail
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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.
 
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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!
 
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.
 
thanks for the reply...as well.
 
LaTeX

iontail said:
...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! :smile:

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 :rolleyes:

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

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