- #1
Zoe-b
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Homework Statement
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 ||
Homework 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.
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