12.8. EXERCISES 321
Is this horrible? Yes it is. However, if you have a computer algebra system do it foryou, it isn’t so bad. For example, to get the last term, you just do
1xx2
x3
( 1 x x2)=
1 x x2
x x2 x3
x2 x3 x4
x3 x4 x5
Then you do the following.
∫ 1
0
1 x x2
x x2 x3
x2 x3 x4
x3 x4 x5
dx =
1 1
213
12
13
14
13
14
15
14
15
16
You could get Matlab to do it for you. Then you add in the last column which consists ofthe original vectors. If you wanted an orthonormal basis, you could divide each vector byits magnitude. This was only painless because I let the computer do all the tedious busywork. However, I think it has independent interest because it gives a formula for a vectorwhich will be orthogonal to a given set of linearly independent vectors.
12.8 Exercises1. Find the best solution to the system
x+2y = 62x− y = 53x+2y = 0
2. Find an orthonormal basis for R3, {w1,w2,w3} given that w1 is a multiple of thevector (1,1,2).
3. Suppose A = AT is a symmetric real n×n matrix which has all positive eigenvalues.Define
(x,y)≡ (Ax,y) .
Show this is an inner product on Rn. What does the Cauchy Schwarz inequality sayin this case?
4. Let ||x||∞≡max
{∣∣x j∣∣ : j = 1,2, · · · ,n
}. Show this is a norm on Cn. Here
x=(
x1 · · · xn
)T.
Show ||x||∞≤ |x| ≡ (x,x)1/2 where the above is the usual inner product on Cn.
5. Let ||x||1 ≡ ∑nj=1
∣∣x j∣∣ .Show this is a norm on Cn. Here x=
(x1 · · · xn
)T.
Show ||x||1 ≥ |x| ≡ (x,x)1/2. where the above is the usual inner product on Cn.Show there cannot exist an inner product such that this norm comes from the innerproduct as described above for inner product spaces.