MATH 203 – 9 JULY 2008 – EXAM 5
Answer each of the following questions. Show all work, as partial credit may be given.
This exam is out of a total of 65 points.
1. (10 pts.) Find a basis for the subspace Sspanned by
3
6
−9
−3
,
−1
−2
3
1
,
6
−2
5
1
and
state the dimension of S.
2. Let A=
−3 6 −1 1 −7
1−2 2 3 −1
2−4 5 8 −4
.
(a) (5 pts.) Find rank(A) and dimN ul(A).
(b) (10 pts.) Find bases for Col(A) and Row(A).
3. (10 pts.) Determine if the set B={t2+ 1,1−t2,1 + t}is a basis for P2, the vector space
of polynomials of degree 2 or less. Fully justify your answer.
4. (10 pts. each) Let B=(" 1
−2#,"5
−6#), and C=(" 3
−2#,"−1
0#), be bases for R2,
and let E=(" 1
0#,"0
1#) be the standard basis.
(a) Find the change of coordinates matrix from Bto the standard basis, and the change
of coordinates matrix from Cto the standard basis.
(b) Find the change of coordinates matrix from Bto Cand from Cto B.
5. (10 pts.) Assuming that λ= 2 is an eigenvalue of the matrix A=
0−4−6
−1 0 −3
125
, find
a basis for the eigenspace of Acorresponding to λ= 2.
