Math 4063-5023
SOLUTIONS TO SECOND EXAM
November 18, 2014
1. Denitions. Write down the precise denitions of the following notions. (3 pts each)
(a) vector space homomorphism (a.k.a. linear transformation)
A vector space homomorphism is a mapping : V W be

Math 4063-5023
SOLUTIONS TO FIRST EXAM
9:00 10:15 am, Oct 2, 2014
You must write your answers in complete sentences and full detail to receive full credit.
1. Denitions. Write down the precise denitions of the following notions. (5 pts each)
(a) subspace

MATH 4063-5023
Homework Set 6
1. Let S be the subspace of R3 spanned by [1, 0, 0] and [0, 1, 0]. Identify let v1 = [1, 1, 3] and let
v2 = [2, 3, 1]. Determine [v1 ]S + [v2 ]S explicitly (it has to be some hyperplane in the direction of S inside
R3 ).
We

MATH 4063-5023
Homework Set 5
1. Suppose that V is a nitely generated vector space and : V W is a linear transformation. Show
that im () W is nitely generated.
Since V is nitely generated it has a nite basis; say, B = cfw_v1 , . . . , vn is a basis for

MATH 4063-5023
Solutions to Homework Set 4
1. Let P be the vector space of polynomials with indeterminant x.Which of the following mappings are
linear transformations from P to itself
(a) T : p xp
Let p1 , p2 be two polynomials and let , F. We have
p1
=

MATH 4063-5023
Solutions to Homework Set 3
1. Test for the solvability of the following linear systems (over R). If the system is solvable, then express
the general solution in the form of x = xp + x0 where x0 is a particular solution of the given linear