Math 304
Quiz 16
Summer 2006
Linear Algebra
1. Use the Gram-Schmidt process to orthonormalize the pair of functions
R1 1and x in the space C[0, 1], where the inner product is given by hf, gi =
f (x)g(x) dx.
0
R1
Solution. Since k1k2 = 0 12 dx = 1, the first function is already
R1
normalized. Now h1, xi = 0 x dx = 21 , so the Gram-Schmidt process
says to replace the function x by the function x − 21 to get a function
orthogonal to the function
1. It remains
this new function.
R1
R 1 2to normalize
1 2
1
1 2
1
Since kx − 2 k = 0 (x − 2 ) dx = 0 (x − x + 4 ) dx = 13 − 21 + 41 = 12
,
the norm kx − 21 k = 2√1 3 . Therefore the final orthonormal pair will be
√
√
1 and 2 3(x − 21 ), or equivalently 1 and 3(2x − 1).
−1 3
, find an orthogonal matrix Q and an upper triangular
2. If A =
1 5
matrix R such that A = QR.
[#2(a), p. 281]
√
Solution. Divide the first column by 2 to normalize it. The scalar
product of the second column with the new first column is − √32 + √52 =
√
√
2. Subtract 2 times
column from the second column to
! thefirst
1
√ −√
4
3
. Normalize the new second column
get
− 2 √1 2 =
4
5
2
√
by dividing it by!4 2. The resulting matrix is the orthogonal matrix
− √12 √12
Q=
. Tracking the operations that were performed shows
√1
√1
2
2
! √
√ √ √
− √12 √12
−1 3
2 √2
2 √2
=
. Thus
.
that R =
√1
√1
1 5
0 4 2
0 4 2
2
2
June 26, 2006
Dr. Boas