1) Define the following: (10 points each)
a) A vector space:
b) The determinant of an n × n Matrix:
c) A linear transformation:
d) An eigenvector:
2) True or False (5 points for a correct answer -5 points for a wrong answer, 0 for not answering.)
T F If A and B are n × n invertible matrices, then (A−1B)
−1 = B−1A.
T F M2,2 has a set of three independent vectors
T F The derivative is a linear transformation from C∞ to C∞.
3) Give examples of the following: (Explain your answers.) (10 points each)
a) Give an example of a nonlinear function from P2(x) to R2
.
b) Give an example of a set of linearly independent vectors in M2,2 which do not span M2,2.
c) Give an example of a linear transformation which is onto but not one-to-one. d) A set of P4(x)
which is not a subspace of P4(x)
4) Determine if the set {3 + 5x + x
2
, −1 + 4x + 3x
2
, 5 − x − 3x
2} is a basis for P2(x). (10 points)
5) Find the determinant of the matrix



1 0 2 0
2 3 0 0
3 0 −2 0
5 3 1 π


. (10 points)
6) Let L : R2 → R2 be a linear transformation whose matrix representation with respect to the standard
basis is the matrix A =

6 4
−2 0 
. (5 points each part)
a) Find the charachteristic polynomial of A.
b) Find the eigenvalues of A.
c) For each eigenvalue, find its eigenspace.
7) Let D : C∞(0, 1) → C∞(0, 1) be a linear map such that D(tan(x)) = sec2
(x), D(erf(x)) = e
−x
2
and
D(x
3
) = 3x
2
. What is D(π tan(x) − 2x
3 + 5erf(x))? (10 points)
8) Let T : P2(x) → R2 be given by T(p(x)) = 
p (0)
p (2) 
.
a) Find the matrix of T with respect to the standard basis for P2(x) which is {1, x, x2} and the standard
basis for R2
. (Hint what is T(x
2
)?) (10 points)
b) Find the matrix representation of T with respect to the basis {x
2 + 1, x + 1, 1} for P2(x) and the basis
 2
1

,

1
1
 for R2
. (10 points)
9) Let T : V → W be a linear transformation, show that ker(T) is a subspace of V . (15 points)