Find a vector $\displaystyle (w_1,w_2,w_3,...)$ that is orthogonal to $\displaystyle v=(1,\frac{1}{2},\frac{1}{4},...)$. Compute its length $\displaystyle ||w||$.

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- May 17th 2011, 07:04 AMalexmahoneOrthogonal vectors
Find a vector $\displaystyle (w_1,w_2,w_3,...)$ that is orthogonal to $\displaystyle v=(1,\frac{1}{2},\frac{1}{4},...)$. Compute its length $\displaystyle ||w||$.

- May 17th 2011, 07:26 AMtonio
- May 17th 2011, 07:29 AMalexmahone
- May 17th 2011, 07:46 AMtonio

There are infinite Hilbert spaces ( to add "with inner product" is futile: ANY Hilbert space is a vector space with an inner product). I can think of at least

two different Hilbert spaces with their inner products to which your vector belongs, so again: what vector space with what iiner product?

Tonio - May 17th 2011, 07:51 AMalexmahone
- May 17th 2011, 08:13 AMtonio

That's what I also thought, but then you already know what to look for...(Nerd)(Giggle)...

Idea: $\displaystyle \sum\limits^\infty_{n=0}\frac{1}{4^n}=\frac{4}{3} \, , \, \, \,\sum\limits^\infty_{n=1}\frac{1}{2^{2n-1}}=2\sum\limits^\infty_{n=1}\frac{1}{4^n}=2\cdot \frac{1}{3}=\frac{2}{3}$

Tonio - May 17th 2011, 08:26 AMalexmahone
- May 17th 2011, 11:54 AMOpalg
- May 17th 2011, 12:27 PMalexmahone