Math310: Sample Exam 2
The following topics are covered by the exam
•Vector spaces: spanning sets, linear independence, basis and dimension, spaces
associated to matrices (column space, row space, nullspace, rank, nullity).
•Linear transformations: definition, kernel and range.
•Coordinate representation of elements of a vector space, change of basis, transi-
tion matrices; matrix representation of linear maps, change of basis formula.
Problem 1. Let Sbe the subspace of R4spanned by
w1=
1
0
2
1
, w2=
1
1
2
0
, w3=
2
1
4
1
, w4=
3
2
6
1
.Let b=
2
0
1
1
.
a) Does bbelong to S. Do w1, w2, w3, w4span R4?
b) Are w1, w2, w3, w4linearly independent?
c) Find a basis of S. What is the dimension of S?
Problem 2. Let A=
1 2 3 −1
3 5 7 1
4 6 10 0
.
a) Find a basis of N(A). What is the dimension of N(A)?
b) What is the rank of A? Will the column vectors of Aspan R3?
c) Will the row vectors of Aspan R4?
d) Are the column vectors of Alinearly independent?
Problem 3. Consider the following vectors in P3
1, x2, x2−2
a) Do they span P3?
b) Are they linearly independent?
c) Find the dimension of Span(1, x2, x2−2).
