Math 314 Final Exam Spring, 1996

Math 309
Sample Final Exam
1.(20 pts) State clearly and concisely the definition of each of the following concepts.
(a) The set of vectors { , ,..., }
n
v1 v2
vn
⊆ R is linearly dependent.
(b) λ is an eigenvalue of the ( nxn) matrix A.
(c)
The matrix
nxn
A∈R is diagonalizable.
(d)
A is row equivalent to B, where A and B are (mxn) matrices.
2.(12 pts) Find a basis for the subspace S P 3
, where S = p∈ P3 p′ (1) = 0 . Here P n
denotes
the vector space of all polynomials of degree less than n.
⊆ { }

⎡1 2 −1 1⎤
3.(24 pts) Given that A =
⎢
2 4 −3 0
⎥
has row echelon form
⎢
⎥
U
⎢⎣1 2 1 5⎥⎦
⎡1 2 −1 1⎤
=
⎢
0 0 1 2
⎥
⎢ ⎥
⎢⎣
0 0 0 0⎥⎦
(a) Find a basis for the null space of A.
(b) Find a basis for the range of A.
(c) Find a basis for the row space of A.
(d) Determine rank ( A)
.
4.(28 pts) True or false. Please explain your answer.

(a) If A = BC where B is nonsingular, then rank( A) = rank( C)
.
4
(b) If A is (4x3) and rank( A) = 3 , then Ax= b is solvable for any b∈
R .
(c)
There exist (2x2) matrices A and B such that both A and B are singular, but
AB is nonsingular.
k
(d) If A is an (nxn) matrix for which A = 0 for some k >1, then A is singular.
5.(20 pts) For each of the following, if U is a subspace of V, verify the subspace criteria. If U is
not a subspace of V, explain precisely why not.
nxn
(a) U = all (nxn) nonsingular matrices, V = R .
nxn
(b) U = all (nxn) symmetric matrices, V = R .

6.(10 pts) Let { x , , 1
xn}
{ x x }
be an orthonormal set in an inner product space V. Prove that
, , 1
n
is linearly independent.
nxn
7.(10 pts) Let A,
B∈R . If A is diagonalizable and B is similar to A, prove that B is also
diagonalizable.
8.(16 pts) Which of the following matrices are diagonalizable and which are not. Please give

oth an answer and a reason.
(a)
⎡2 2 1 ⎤
⎢
0 1 2
⎥
⎢ ⎥
⎢⎣0 0 −1⎥⎦
(b)
⎡2 −1⎤
⎢
0 2
⎥
⎣ ⎦
3
9.(16 pts) Consider the linear transformation L:
P → P
4 defined by L( p)( x) = xp( x)
. Find the
matrix which represents L relative to the bases { 2
2 3
1,1 −x,1
− x+ x } in P
3
and { 1, x, x , x } in P 4
.

10.(20 pts) Given the data, find the equation of the line y= a0 + a1x
which best fits the data in
the least squares sense.
x 0 3 6
y 1 4 5
⎡1 1 1⎤
11.(14 pts) Given A =
⎢
1 0 1
⎥
, factor A into a product QR, where Q is orthogonal and R is
⎢⎢ ⎥
⎣0 1 1 ⎥ ⎦
upper triangular and nonsingular.

⎡2 −3⎤
12.(10 pts) The matrix A = ⎢ has eigenvalues
⎣2 −5
⎥ λ
1
= 1 and λ
2
= − 4 . Find the
⎦
corresponding eigenvectors.