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Higher linear algebra

  1. Sep 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Hi - i have fully worked solutions in my notes, but i do not understand this step in the proof. Proposition 3.16. Suppose that W is a subspace of a nite-dimensional inner-product
    space V and let v be an alement of V . Then ||v - projW(v)|| <= ||v - w|| for all w is an element of W. Moreover, if
    w element of W and ||v - projW(v)|| = ||v - w|| then projW(v) = w.
    Proof. If w is an element of W then
    ||v - projW(v)||^2 + ||v - projW(v)||^2 + || projW(v) - w||^2 (1)
    = ||v - projW(v) + projW(v) - w||^2 (by Pythagoras' Theorem) (2)
    = ||v - w||^2:

    note - projW(v) is the projection of v onto W

    I do not really understand how (1) implies (2)? Thanks for ur help!




    3. The attempt at a solution
     
  2. jcsd
  3. Sep 27, 2009 #2
    There is a typo in (1).

    ||v - projW(v)||^2 + ||v - projW(v)||^2 + || projW(v) - w||^2 (1)

    should be

    ||v - projW(v)||^2 + || projW(v) - w||^2 (1).


    Now Pythagoras says if vectors a and b are orthogonal, then

    ||a||^2 + ||b||^2 = ||a + b||^2.
     
  4. Sep 27, 2009 #3
    thanks!
     
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