(PLEASE SEE ATTACHMENT FOR FULL QUESTION) Let V be a nite dimensional inner product space, and let W be a subspace of V. Prove the following...

(PLEASE SEE ATTACHMENT FOR FULL QUESTION)

Let V be a nite dimensional inner product space, and let W be a subspace of V. Prove

the following statements.

(a) For all ~x; ~y 2 V, h~x; ~yi = hprojW(~x); projW(~y)i + hperpW(~x); perpW(~y)i.

(b) Suppose ~v 2 V. There exists a vector ~x 2 W such that h~v; ~xi = 1 if and only if

~v =2 W?.

(c) If dimW = k and dimV = n, then then exists a basis B for V such that

B[projW]B =

~e1 ~e2 : : : ~ek ~0 ~0 : : : ~0

;

(The rst k columns of the matrix are standard basis vectors ~e1; : : : ; ~ek 2 Rn, and

the last n ???? k columns of the matrix are ~0 2 Rn.)