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!!