Let V be an inner product space. Prove that if v and w are vectors in V such that <v,x> = <w,x> for every x in V, then v = w.

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- April 16th 2010, 06:56 PMmathbugInner Products
Let V be an inner product space. Prove that if v and w are vectors in V such that <v,x> = <w,x> for every x in V, then v = w.

- April 16th 2010, 07:07 PMDefunkt
- April 16th 2010, 07:53 PMmathbug
So then <v,x> = sum <v,vi> [<vi,x>] = sum <w,vj> [<vj,x>] = <w,x> Then

<v,v1>[<v1,x>] - <w,v1>[<v1,x>] +..+ <v,vn>[<vn,x>] - <w,vn>[<vn,x>]=0

So then where am I supposed to go from here? - April 17th 2010, 12:50 AMOpalg
- April 17th 2010, 03:07 AMDefunkt
- April 17th 2010, 10:44 AMmathbug
But how can you say that <v,v_i> = <w,v_i> for all x in V? If we had

<v,v1>[<v1,x>] - <w,v1>[<v1,x>] +..+ <v,vn>[<vn,x>] - <w,vn>[<vn,x>]=0

then we could factor and get

[<v1,x>][<v,v1>-<w,vi>] +...+ [<vn,x>][<v,vn>-<w,vn>]=0

I get that if <vi,x> were linearly independent, we would have that <w,v_i>=<v,v_i> for all x in V, but how can we show that? - April 17th 2010, 03:08 PMDefunkt
- April 17th 2010, 09:34 PMmathbug
Oh! You're right, I do tend to make things too complicated. That answer is very simple! But makes so much sense... I completely understand now. Thank you so much for all your help!