MATH 4063-5023
Homework Set 5
1. Find the
20
(a) 0 2
00
characteristic polynomials and minimal polynomials of the following matrices.
0
0
2
The characteristic polynomial of this matrix is
2
0
0
2
0 = (2 )3
pA () = det 0
0
0
2
The minimal polynomial will
MATH 4063-5023
Solutions to Homework Set 3
1. Let F be a eld with exactly two elements (it will be isomorphic to Z2 ) and let V be a 2-dimensional
vector space over F. How many vectors are there in V ? How many dierent bases are there for V ?
Since any e
MATH 4063-5023
Solutions to Homework Set 1
1. Let F be a eld, and let Fn denote the set of n-tuples of elements of F, with operations of scalar
multiplication and vector addition dened by
for all F and all [1 , . . . , n ] in Fn
[1 , . . . , n ]
:
= [a1
MATH 4063-5023
Solutions to Homework Set 6
1. Determine if the following matrices A are diagonalizable. When A is diagonalizable, provide both the
matrix C diagonalizing A and its diagonal form D (so that D = C1 AC).
(a) A =
2
0
1
1
We have
det (A I) = (
LECTURE 1
Introduction
The rudiments of linear algebra are known to every scientist who knows what a vector is and every software
engineer who knows what an array is. In Math 3013 (Linear Algebra) these rudimentary ideas are abstracted
and generalized so
LECTURE 5
Finitely Generated Vector Spaces
We are now in position to prove some general theorems about nite dimensional vector space that will be
crucial to a number of applications.
But before starting on this, let me try to explain again, in a dierent w
LECTURE 4
Elementary Operations and Matrices
In the last lecture we developed a procedure for simplying the set of generators for a given subspace of the
form
S = spanF (v1 , . . . , vk ) := cfw_1 v1 + + k vk | 1 , . . . , k F
It went like this
Find a de
LECTURE 3
Dimension and Bases
In the preceding lecture, we introduced the notion of a subspace of a vector space and an easy way to
construct subspace; namely, by considering the set of all possible linear combinations of a set of vectors: if
cfw_v1 , v2
LECTURE 2
Subspaces and Linear Independence
Last time we dened the notion of a eld F as a generalization of the set of real numbers, and the notion of
a vector space over a eld F as a generalization of the vector space Rn (or any other vector space studie
Math 4063-5023
SOLUTIONS TO FIRST EXAM
2:30 3:20 , October 23, 2013
1. Denitions. Write down the precise denitions of the following notions. (3 pts each)
(a) dependence relation.
A dependence relation amongst a set of vectors cfw_v1 , . . . , vk in a ve
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
=
LECTURE 6
Quotient Spaces
In all the development above we have created examples of vector spaces primarily as subspaces of other
vector spaces. Below we provide a construction which starts with a vector space V over a eld F and
ll
a subspace S of V , and
LECTURE 7
Systems of Linear Equations
You may recall that in Math 3013, matrices were rst introduced as a means of encapsulating the essential
data underlying a system of linear equations; that is to say, given a set of n linear equations in m variables
a
LECTURE 12
Linear Transformations and Matrices
1. Matrix Multiplication
We have put this o long enough; let us now dene matrix multiplication.
Let M atn,m (F) be the set of n m matrices with entries in F.
of M atn,m (F) as an array with n rows and m colum
LECTURE 10
Hyperplanes
Let V be a nitely generated vector space over a eld F. Today we will focus our attention on a special
class of subsets of V . These subsets will not in general be subspaces, but they arise quite naturally in linear
algebra and have
LECTURE 9
Homogeneous Linear Systems
We now return to some more theoretical aspects linear systems and their corresponding matrices.
We rst note that there is a natural 1:1 correspondence between homogeneous n m linear systems and
n m matrices. For any n
LECTURE 8
Solving Linear Systems
The basic method for solving linear systems that taught in Math 3013 is based on the notion of a matrix
in reduced row echelon form.
Let A be an n m matrix with entries in a eld F. Recall that the rst (reading left to righ
LECTURE 13
Homomorphisms and Isomorphisms
While I have discarded some of Curtiss terminology (e.g. linear manifold) because it served mainly to
reference something (dierential geometry) that is esoteric to the present course; I now nd myself wanting
to br
LECTURE 14
Endomorphisms and Automorphisms
We now specialize to the situation where a vector space homomorphism (a.k.a, linear transformation) maps
a vector space to itself.
Definition 14.1. Let V be a vector space over a eld F. A vector space homomorphis
LECTURE 18
The Cayley-Hamilton Theorem and the Jordan Decomposition
Let me begin by summarizing the main results of the last lecture. Suppose T is a endomorphism of a
vector space V . Then T has a minimal polynomial that factorizes into a product of power
LECTURE 17
Invariant Subspaces
Recall the range of a linear transformation T : V W is the set
range (T ) = cfw_w W | w = T (v ) for some v V
Sometimes we say range (T ) is the image of V by T to communicate the same idea. We can also generalize
this noti
LECTURE 16
The Theory of a Single Endomorphism
Recall that an endomorphism is a map T : V W between two vector spaces that is compatible with the
two vector space operations
(i) T (v) = T (v) for all F and for all v V .
(ii) T (v = v ) = T (v) + T (v ) fo
LECTURE 15
Determinants
We are now going to break with the text in two ways. First of all, since we have just spent the last couple
weeks discussing applications of matrices to linear algebra, to me it seems much more natural to move
directly to the theor
LECTURE 11
Linear Transformations
1. Functions Between Sets
Although I think it safe to assume that everybody here is familiar with the utility of functions, it may not
s
be the case that everybody keeps in mind the generality of this concept. So let me t