Orthogonal projection/norm

1. Apr 30, 2012

Zoe-b

1. The problem statement, all variables and given/known data
Let U be the orthogonal complement of a subspace W of a real inner product space V.
Have already shown that T is a projection along a subspace W onto U, and that V is the direct sum of W and U.

The questions now says: show
||T(v)|| = inf (w in W) || v - w ||

2. Relevant equations
I have some vague notion that in R^3 say, an orthogonal projection can be used to find the shortest distance between a plane and a point. I have absolutely no idea how to prove this using inner products though.

3. The attempt at a solution

if we write T' for the projection along U onto W, then we have:

v = (T + T')(v)

T(v) = v - T'(v)

now T'(v) is in W, but I don't know how to show it is the w that minimises || v - w ||

Any suggestions for resources would also be welcome- this is not in my notes at all and google hasn't been that helpful :P

2. May 2, 2012

susskind_leon

There is a theorem called 'projection theorem' which gives you exactly that, but it only works in Hilbert spaces, so I'm not sure if it's general enough.

3. May 2, 2012

micromass

Staff Emeritus
Let $w_0=v-T(v)$, then w is supposed to be the point in W which is closest to v. Can you prove that for each w in W holds that

$$\|v-w_0\|\leq \|v-w\|$$

Hint: draw a picture and see if you get something perpendicular. Use Pythagoras theorem.

4. May 4, 2012

Zoe-b

Thanks for the replies- Hilbert spaces aren't on my course yet so I don't think thats the way to go. I'm trying to use the second hint and do it with a diagram but not getting that far- also this is in any real inner product space not necessarily R^n..

5. May 4, 2012

micromass

Staff Emeritus
T(v) is orthogonal to W, right??

So v-w0, w-w0 and v-w forms a right triangle.

6. May 4, 2012

Zoe-b

Yeah done it now, thanks! I guess I did use pythagoras- Just conceptually not that happy with drawing triangles when the elements I'm using aren't necessarily vectors? Anyway, done it using just inner product notation and now happy :)