1. ## independant proof

there is $\{y_{1}..y_{n}\}$ a group of vectors in $R^{n}$ ,there is $\{u_{1}..u_{n}\}$ orthonormal basis of $R^{n}$.
the first part i have proved that for $\lambda_{1}..\lambda_{n}\$ in R
$\parallel\sum_{i=1}^{n}\lambda_{i}y_{i}\parallel^{ 2}\leq(\sum_{i=1}^{n}|\lambda_{i}|^{2})(\sum_{i=1} ^{n}\parallel y_{i}\parallel^{2})$
the second part is my question
prove that if $\{y_{1}..y_{n}\}$ is a set for which $\sum_{i=1}^{n}\parallel y_{i}\parallel^{2}<1$then $\{y_{1}+u_{1,}..,y_{n}+u_{n}\}$ is independant?
how i tried to solve it:
a set is independant if for $k_{1}(y_{1}+u_{1})+..+k_{n}(y_{n}+u_{n})=0$the only way it to be zero is if k1=..=kn=0
i opened it and got $k_{1}y_{1}+..+k_{n}y_{n}=k_{1}u_{1}+..+k_{n}u_{n}$
so on both side we have equal vectors so is their norm
$(\sum_{i=1}^{n}|k_{i}|^{2})(\sum_{i=1}^{n}\paralle l y_{i}\parallel^{2})=(\sum_{i=1}^{n}|k_{i}|^{2})(\s um_{i=1}^{n}\parallel u_{i}\parallel^{2})
$
what next how to prove that $k_{1}=..=k_{n}=0$
??

2. ## Re: independant proof

We have $\sum_{i=1}^{n}k_iy_i=-\sum_{i=1}^{n}k_iu_i$ . Taking norms we obtain $||{\sum_{i=1}^n{k_iy_i}}||^2=\sum_{i=1}^n|k_i|^2$ . But $\sum_{i=1}^n|k_i|^2\leq (\sum_{i=1}^n|k_i|^2)(\sum_{i=1}^n||y_i||^2)$ . As $\sum_{i=1}^n||y_i||^2\right$< $1$ , necessarily $\sum_{i=1}^n|k_i|^2=0$ which implies $k_i=0$ for all $i=1,\ldots,n$ .

3. ## Re: independant proof

Originally Posted by FernandoRevilla
We have $\sum_{i=1}^{n}k_iy_i=-\sum_{i=1}^{n}k_iu_i$ . Taking norms we obtain $||{\sum_{i=1}^n{k_iy_i}}||^2=\sum_{i=1}^n|k_i|^2$ . But $\sum_{i=1}^n|k_i|^2\leq (\sum_{i=1}^n|k_i|^2)(\sum_{i=1}^n||y_i||^2)$ . As $\sum_{i=1}^n||y_i||^2\right$< $1$ , necessarily $\sum_{i=1}^n|k_i|^2=0$ which implies $k_i=0$ for all $i=1,\ldots,n$ .
why if orthonormal then $||{\sum_{i=1}^n{k_iy_i}}||^2=\sum_{i=1}^n|k_i|^2$
?

norm of one vector is 1
but here we have a sum

4. ## Re: independant proof

Originally Posted by transgalactic
why if orthonormal then $||{\sum_{i=1}^n{k_iy_i}}||^2=\sum_{i=1}^n|k_i|^2$ norm of one vector is 1 but here we have a sum
If $\{u_i:i=1,\ldots,n\}$ is an orthonormal system then, $||{\sum_{i=1}^n{k_iu_i}}||^2 =<\sum_{i=1}^n{k_iu_i},\sum_{i=1}^n{k_iu_i}>=...$

Conclude.

5. ## Re: independant proof

Originally Posted by FernandoRevilla
If $\{u_i:i=1,\ldots,n\}$ is an orthonormal system then, $||{\sum_{i=1}^n{k_iu_i}}||^2 =<\sum_{i=1}^n{k_iu_i},\sum_{i=1}^n{k_iu_i}>=...$

Conclude.
$<\sum_{i=1}^n{u_i},\sum_{i=1}^n{u_i}>$
again we have here a sum of vectors not a single vector,
the norm of the single vector is 1

why the resolt is 1 too?