http://i97.photobucket.com/albums/l2...7/mathrev1.jpg

Really stuck on this, I know that two vectors are orthogonal if their inner product space is 0, and the're orthonormal if they have norm 1, but i'm really not sure how to go about this question.

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- May 12th 2009, 06:15 AMpkrOrthonormal vectors
http://i97.photobucket.com/albums/l2...7/mathrev1.jpg

Really stuck on this, I know that two vectors are orthogonal if their inner product space is 0, and the're orthonormal if they have norm 1, but i'm really not sure how to go about this question. - May 12th 2009, 07:33 AMHallsofIvy
Hopefully, you also know some basic properties of the inner product:

<u+ v, w>= <u,w>+ <v,w>

<au, v>= a<u, v>

and

$\displaystyle <u, v>= \overline{<v, u>}$

if the vector space is over the complex numbers (the overline is the complex conjugate).

Using those, $\displaystyle <av_1+ bv_2, cv_1+ dv_2>= <av_1,cv_1+ dv_2>+ <bv_2,cv_1+ dv_2>$$\displaystyle = <av_1,cv_1>+ <av_1,dv_2>+ <bv_2,cv_1>+ <bv_2, cv_1>$$\displaystyle = a\overline{c}<v_1,v_1>+ a\overline{d}<v_1,v_2>+ b\overline{c}<v_2,v_1>+ b\overline{d}<v_2,v_1>$ and now you can use the fact that $\displaystyle v_1$ and $\displaystyle v_2$ are orthonormal.

Apply that to your problem. Tedious but straightforward.

(I just re-read the problem and noticed that the set of "v" vectors in the definition of w1 and w2 are disjoint! Gosh this problem is**trivial**!)