1. ## Orthogonal 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 $\displaystyle ||w||$.

2. Originally Posted by alexmahone
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 $\displaystyle ||w||$.

Orthogonal...in what vector space and with respect to what inner product??

Tonio

3. Originally Posted by tonio
Orthogonal...in what vector space and with respect to what inner product??

Tonio
Hilbert space with respect to dot product.

4. Originally Posted by alexmahone
Hilbert space with respect to dot product.

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

5. Originally Posted by tonio
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
I think the question merely requires us to find $\displaystyle w_1,w_2,w_3,...$ such that $\displaystyle w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0$.

6. Originally Posted by alexmahone
I think the question merely requires us to find $\displaystyle w_1,w_2,w_3,...$ such that $\displaystyle w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0$.

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

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

7. Originally Posted by tonio
That's what I also thought, but then you already know what to look for......

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
I think the question merely requires us to find $\displaystyle w_1,w_2,w_3,...$ such that $\displaystyle w_1+\frac{w_2}{2}+\frac{w_3}{4}+...=0$.
You also need to ensure that $\displaystyle \sum|w_n|^2<\infty$, and in fact you need to be able to compute that sum in order to do the second part of the question.
You also need to ensure that $\displaystyle \sum|w_n|^2<\infty$, and in fact you need to be able to compute that sum in order to do the second part of the question.
Thanks. $\displaystyle w=(1,-2,0,...)$ and its length is $\displaystyle \sqrt{5}$.