1. ## Orthogonal Complements

This should be easy, I'm not sure what I'm missing!! Any help would be great...

Let W be a finite dim subspace of an inner product space V. Show $\displaystyle \exists$ a projection, $\displaystyle T$ on $\displaystyle W$ along $\displaystyle W^\perp$ such that the nullspace of T, $\displaystyle N(T)=W^\perp$.
Then show $\displaystyle ||T(x)||\leq ||x|| ~~ \forall x\in V$

What I know:
So far, I know that by a theorem, there are unique vectors $\displaystyle u\in W, z\in W^\perp$ such that $\displaystyle y=u+z$

I know by definition $\displaystyle T:V\rightarrow V$ is a projection on W along $\displaystyle W^\perp$ if for $\displaystyle x=x_1+x_2$ with $\displaystyle x_1\in W, x_2\in W^\perp$, we have $\displaystyle T(x)=x_1$

Finally, I know if x and y are orthogonal vectors, that $\displaystyle ||x+y||^2 =||x||^2+||y||^2$

It seems like it should fall into place so easily... I just don't know how to define my T function so it works for each case and still lands me in the complement of W. Please help!!

2. Originally Posted by MatthewD
This should be easy, I'm not sure what I'm missing!! Any help would be great...

Let W be a finite dim subspace of an inner product space V. Show $\displaystyle \exists$ a projection, $\displaystyle T$ on $\displaystyle W$ along $\displaystyle W^\perp$ such that the nullspace of T, $\displaystyle N(T)=W^\perp$.
Then show $\displaystyle ||T(x)||\leq ||x|| ~~ \forall x\in V$

What I know:
So far, I know that by a theorem, there are unique vectors $\displaystyle u\in W, z\in W^\perp$ such that $\displaystyle y=u+z$

I know by definition $\displaystyle T:V\rightarrow V$ is a projection on W along $\displaystyle W^\perp$ if for $\displaystyle x=x_1+x_2$ with $\displaystyle x_1\in W, x_2\in W^\perp$, we have $\displaystyle T(x)=x_1$

Finally, I know if x and y are orthogonal vectors, that $\displaystyle ||x+y||^2 =||x||^2+||y||^2$

It seems like it should fall into place so easily... I just don't know how to define my T function so it works for each case and still lands me in the complement of W. Please help!!
well, you've already defined $\displaystyle T$. for the second part, if $\displaystyle x=x_1+x_2, \ x_1 \in W, \ x_2 \in W^{\perp},$ then $\displaystyle ||T(x)||^2=||x_1||^2 \leq ||x_1||^2+||x_2||^2=||x_1+x_2||^2=||x||^2.$

(note that since $\displaystyle \langle x_1,x_2 \rangle = \langle x_2,x_1 \rangle = 0,$ we have $\displaystyle ||x_1+x_2||^2=\langle x_1+x_2, x_1+x_2 \rangle=||x_1||^2+||x_2||^2.$)