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MATH 2150 HW 3
DUE: Thursday, April 30th, 2009
0 6 6 3
1 2 1 1 

1. Let A = 
 4 1  3 4


1 3 2 0 
Find
a. a basis for the row space of A
b. a basis for the column space of A
c. a basis for the nullspace of A
d. rank of A
e. nullity of A
2. Let V = C[-1, 1] = vector space of all continuous functions defined on [-1, 1] with the standard
definitions of addition and scalar multiplication. We define an inner product on V as
1
<f, g> =
 f ( x) g ( x)dx
1
Find a.
<f, g>
b.
2
f(x) = -x and g(x) = x - x + 2
||g||
c.
d(f, g)
where
3. Let W be the subspace of R 4 (with Euclidean inner product) spanned by the vectors (1, 1, 0, 7) and
(2, 1, 2, 6). Find a basis for the orthogonal complement of W.
3. Let u = (1, -3) and v = (5, 1). Use the inner product <u, v> = 3u 1 v 1 + 5u 2 v 2 to compute the following:
a. || v ||
b.
d(u, v)
c. the angle between u and v.
4. Suppose that u, v, and w are vectors in an inner product space such that
<u., v> = 5, <v, w> = -2, <u, w> = 1, || u || = 2, || v || = 1, || w || = 3. Evaluate the given expression.
a. || u + v || b. <2u + 3v, 4v – w>
5. Let V = R 2 and let u = (u 1 , u 2 ) and v = (v 1 , v 2 ) be in R 2 . We define <u, v> as
<u, v> = u 1 u 2 + v 1 v 2 (*)
Does (*) define an inner product on V?
6. Let u = (2, -2, 1). Find the coordinate vector of u relative to the orthonormal basis
1
3
3
1
B = {(
, 0,
), (0, 1, 0), (
, 0,
)} using the Euclidean inner product on R 3 .
10
10
10
10
7. Consider R 3 with the Euclidean inner product. In each part, use the Gram Schmidt process to transform
the given basis into an orthonormal basis.
(a) {(1, 1, 1), (1, 1, 0), (1, 0 , 0)}
(b) {(1, -2, 2), (2, 2, 1), (2, -1, -2)}
4 0 1 
8. Let A = 2 3 2


1 0 4
a. Find eigenvalues of A.
b. For each eigenvalue of A, find a basis for the eigenspace of A.
c. Is A diagonalizable? If yes, find a matrix P that diagonalizes A and then determine P 1 AP.