Understanding Projection in Finite-Dimensional Inner-Product Spaces

[Butelle]

Homework Statement

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!

The Attempt at a Solution

 

[Billy Bob]

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.

 

[Butelle]

thanks!