Math 304
Quiz 15
Summer 2007
Linear Algebra
1. Find values of
1
the matrix
2
3
is an eigenvector of
a and b for which the vector
4
a
with eigenvalue 5.
b
Solution. The given information says that
1 a
3
3
=5
.
2 b
4
4
Equivalently,
3 + 4a = 15,
6 + 4b = 20.
Solving each of these equations shows that a = 3 and b = 7/2.
R1
2. In the space C[0, 1] with inner product hf, gi = 0 f (x)g(x) dx, use the
Gram-Schmidt procedure to find an orthonormal basis for the subspace
spanned by the functions 1 and x.
R1
Solution. Since k1k2 = h1, 1i = 0 12 dx = 1, the function 1 is alR1
ready normalized. Now hx, 1i = 0 x dx = 21 , so the function x − 12
is orthogonal to the function 1. It remains to normalize the function
x − 12 .
R1
R1
1
Since kx − 12 k2 = 0 (x − 21 )2 dx = 0 (x2 − x + 41 ) dx = 31 − 21 + 41 = 12
,
√
√
1
the normalized function equals 12 (x − 2 ), or 3 (2x − 1).
√
Thus the required orthonormal basis is [1, 3 (2x − 1)].
June 26, 2007
Dr. Boas