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Homework Help: Orthogonal projection/norm

  1. Apr 30, 2012 #1
    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. jcsd
  3. May 2, 2012 #2
    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.
  4. May 2, 2012 #3
    Let [itex]w_0=v-T(v)[/itex], 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

    [tex]\|v-w_0\|\leq \|v-w\|[/tex]

    Hint: draw a picture and see if you get something perpendicular. Use Pythagoras theorem.
  5. May 4, 2012 #4
    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..
  6. May 4, 2012 #5
    T(v) is orthogonal to W, right??

    So v-w0, w-w0 and v-w forms a right triangle.
  7. May 4, 2012 #6
    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 :)
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