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1
МИНИCTEPCTBO ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ
ФЕДЕРАЛЬНОЕ ГОСУДАРСТВЕННОЕ АВТОНОМНОЕ ОБРАЗОВАТЕЛЬНОЕ
УЧРЕЖДЕНИЕ ВЫСШЕГО ПРОФЕССИОНАЛЬНОГО ОБРАЗОВАНИЯ
«СЕВЕРО-КАВКАЗСКИЙ ФЕДЕРАЛЬНЫЙ УНИВЕРСИТЕТ»
MATHEMATICS
(МАТЕМАТИКА)
УЧЕБНОЕ ПОСОБИЕ
Направление подготовки 21.03.01 – Нефтегазовое дело
Бакалавриат
Ставрополь
2015
2
УДК 51 (075.8)
М 34
ББК 22.1 я73
М 34
MATHEMATICS (Математика): учебное пособие / авт.-сост.:
Н. В. Ширяева, А.С. Мараховский. – Ставрополь: Изд-во СКФУ,
2015. – 236 с.
Пособие составлено в соответствии с требованиями ФГОС ВПО и
предназначено для обучения иностранных студентов основам математики.
Пособие включает три раздела: курс лекций, практикум и методические
рекомендации по организации самостоятельной работы.
Рекомендовано для иностранных студентов, изучающих математику на
английском языке.
Авторы-составители:
канд. психол. наук, доцент Н. В. Ширяева,
канд. физ-мат. наук, д-р. экон. наук, доцент А. С. Мараховский
Рецензенты:
канд. физ-мат. наук, профессор А. С. Адамчук,
д-р. экон. наук, профессор Е. Л. Торопцев
редакционно-издательского совета
Северо-Кавказского федерального
© ФГАОУ ВПО «Северо-Кавказский
федеральный университет», 2014
Печатается по решению
университета
УДК 51 (075.8)
ББК 22.1 я73
3
СHAPTER 1
312
517
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3
5
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Section 1. Linear algebra
and analytical geometry
Topic 1. Matrix. Matrix Operations
Introduction. This lecture is devoted to the discussion of the following concepts: matrix, matrix equality, matrix operations, transposes and
symmetric matrices.
We begin with a definition of a totally general set of matrices, and see
where that takes us.
A matrix is a rectangular array of numbers. Several of them are re-
ferred to as matrices. For example, here is a matrix
Definition VSM: Vector Space of m×n Matrices. The vector space
M(m×n) is the set of all m×n matrices with entries from the set of com-
plex numbers.
Just as we made, and used, a careful definition of equality for column
vectors, so too, we have precise definitions for matrices.
The size or dimension of a matrix is defined as m × n where m is the
number of rows and n is the number of columns. The above matrix is a 3
× 4 matrix because there are two rows and three columns.
The first row is (3 -1 2) , the second row is (5 -1 7) The first column
-
-
.
is
.
When specifying the size of a matrix, you always list the number of rows
before the number of columns. Also, you can remember the columns are like
columns in a Greek temple. They stand up right while the rows just lay there
like rows made by a tractor in a plowed field. Elements of the matrix are identified according to position in the matrix. For example, 7 is in position 2, 3
because it is in the second row and the third column.
Using this notation on the above matrix, a23 = 7.
4
Definition ME: Matrix Equality. The m×n matrices A and B are
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equal, written A=B provided [A] ij=[B] ij for all 1≤i≤m, 1≤j≤n.
So equality of matrices translates to the equality of complex numbers,
on an entry-by-entry basis. Notice that we now have yet another definition that uses the symbol “=” for shorthand. Whenever a theorem has a
conclusion saying two matrices are equal (think about your objects), we
will consider appealing to this definition as a way of formulating the toplevel structure of the proof.
Two matrices are equal exactly when they are the same size and the
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,
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because they are
corresponding entries are identical. Thus
different sizes.
We will now define two operations on the set M(mn). Again, we will
overload a symbol (`+') and a convention (juxtaposition for scalar multiplication).
Definition MA: Matrix Addition. Given the m×n matrices A and B,
define the sum of A and B as an m×n matrix, written A+B, according to
[A+B] ij= [A] ij+[B] ij, 1≤i≤m,1≤j≤n
So matrix addition takes two matrices of the same size and combines
them (in a natural way!) to create a new matrix of the same size. Perhaps
this is the “obvious” thing to do, but it does not relieve us from the obligation to state it carefully.
Example 1. Addition of two matrices
521
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ABC
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412
Example 2.
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=
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,
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=
BA
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=-= ABX
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5
Our second operation takes two objects of different types, specifically
a number and a matrix, and combines them to create another matrix. As
with vectors, in this context we call a number a scalar in order to emphasize that it is not a matrix.
Definition MSM: Matrix Scalar Multiplication. Given the m×n
matrix A and the scalar α∈C, the scalar multiple of A is an m×n matrix,
written αA and defined according to [αA]
Notice again that we have yet another kind of multiplication, and it is
=α[A] ij 1≤i≤m,1≤j≤n
ij
again written putting two symbols side-by-side. Computationally, scalar
matrix multiplication is very easy.
With definitions of matrix addition and scalar multiplication we can
now state, and prove, several properties of each operation, and some
properties that involve their interplay. We now collect ten of them here
for later reference.
Theorem VSPM: Vector Space Properties of Matrices. Suppose
that M
and scalar multiplication as defined in Definition MA and Definition
is the set of all m×n matrices (Definition VSM) with addition
mn
MSM. Then
1. ACM Additive Closure, Matrices
If A,B∈ Mmn, then A+B∈ Mmn.
2. SCM Scalar Closure, Matrices
If α∈C and A∈ Mmn, then αA∈ Mmn.
3. CM Commutativity, Matrices
If A,B∈ Mmn, then A+B=B+A.
4. AAM Additive Associativity, Matrices
If A,B,C∈ Mmn, then A+(B+C)=(A+B)+C.
5. ZM Zero Matrix, Matrices
There is a matrix, 0, called the zero matrix, such that
A+0=A for all A∈ Mmn.
6. AIM Additive Inverses, Matrices
If A∈Mmn, then there exists a matrix −A∈ Mmn so that
A+(−A)=0.
7. SMAM Scalar Multiplication Associativity, Matrices
6
If α,β∈C and A∈ Mmn, then α(βA)=(αβ)A.
8. DMAM Distributivity across Matrix Addition, Matrices
If α∈C and A,B∈ Mmn, then α(A+B)=αA+αB.
9. DSAM Distributivity across Scalar Addition, Matrices
If α,β∈C and A∈ Mmn, then (α+β)A=αA+βA.
10. OM One, Matrices
If A∈ Mmn, then 1A=A.
To prove Property DSAM, (α+β)A=αA+βA, we need to establish the
equality of two matrices (see Proof Technique GS). Definition ME says
we need to establish the equality of their entries, one-by-one. How do we
do this, when we do not even know how many entries the two matrices
might have? This is where the notation for matrix entries, given in Definition M, comes into play. Ready? Here we go.
For any i and j, 1≤i≤m, 1≤j≤n, [(α+β)A] ij Definition MSM
=(α+β)[A]ij =α[A] ij +β[A] ij Distributivity in C
=[αA] ij +[βA] ij Definition MSM
=[αA+βA] ij Definition MA
There are several things to notice here. (1) Each equals sign is an
equality of scalars (numbers). (2) The two ends of the equation, being
true for any i and j, allow us to conclude the equality of the matrices by
Definition ME. (3) There are several plus signs, and several instances of
juxtaposition. Identify each one, and state exactly what operation is being represented by each.
The zero matrix described in this theorem, 0, is what you would expect — a matrix full of zeros.
Definition ZM: Zero Matrix. The m×n zero matrix is written as 0=
0
and defined by [0]
m×n
=0, for all 1≤i≤m, 1≤j≤n.
ij
Subsection TSM: Transposes and Symmetric Matrices
We describe one more common operation we can perform on matrices. Informally, to transpose a matrix is to build a new matrix by swapping its rows and columns.
Definition TM: Transpose of a Matrix. Given an m×n matrix A, its
transpose is the n×m matrix A
t
given by [At]ij =[A] ij,1≤i≤n,1≤j≤m
7
Example TM
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A
Suppose
=
.
We could formulate the transpose, entry-by-entry, using the definition. But it is easier to just systematically rewrite rows as columns (or
vice-versa). The form of the definition given will be more useful in
proofs. So we have
t
A
=
It will sometimes happen that a matrix is equal to its transpose. In this
case, we will call a matrix symmetric. These matrices occur naturally in
certain situations, and also have some nice properties, so it is worth stating the definition carefully. Informally a matrix is symmetric if we can
“flip” it about the main diagonal (upper-left corner, running down to the
lower-right corner) and have it look unchanged.
Definition SYM: Symmetric Matrix. The matrix A is symmetric if
A= At
Before reading the next proof, we want to offer you some advice
about how to become more proficient at constructing proofs. Perhaps
you can apply this advice to the next theorem.
Theorem SMS: Symmetric Matrices are Square. Suppose that A is
a symmetric matrix. Then A is square.
We finish this section with three easy theorems, but they illustrate the
interplay of our three new operations, our new notation, and the techniques used to prove matrix equalities.
Theorem TMA: Transpose and Matrix Addition. Suppose that A
and B are m×n matrices. Then (A+B)t =At+Bt.
Theorem TMSM: Transpose and Matrix Scalar Multiplication.
Suppose that α∈C and A is an m×n matrix. Then (αA)t=αAt.
Theorem TT: Transpose of a Transpose. Suppose that A is an m×n
matrix. Then (At)t=A.
8
Reading Questions.
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1. Perform the following matrix computation:
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ø
2. Let А=
, В=
, С=
Let α = 2 Perform
-
the following calculations(1) A+B, (2) A+C,(3)α B
3. Solve the given matrix equation for α, or explain why no solution
exists:
a
+=
-
.
Topic 2. Matrix Multiplication
Introduction. This lecture is devoted to the discussion of the following concepts: a row vector, a column vector matrix multiplication, Scalar
Matrix Multiplication. For a full understanding of the topic you have an
idea about matrices.
Definition Matrices which are n × 1 or 1 × n are called vectors and
are often denoted by a bold letter. Thus the n × 1 matrix
=
is
also called a column vector. The 1 × n matrix
a row vector.
In more general terms,
12
is called
9
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()()()
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ij
AA
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()
ij
BB
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3
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++
++
Thus you take x1 times the first column, add to x2 times the second
column, and finally x3 times the third column. The above sum is called a
linear combination of the given column vectors.
For A and B matrices, in order to form the product, AB the number of
columns of A must equal the number of rows of B.
Note the two outside numbers give the size of the product. Remember:
If the two middle numbers don’t match, you can’t multiply the
matrices!
Multiply the following
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134
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First check if it is possible. This is of the form (3 × 3) (2 × 3) . The
inside numbers do not match and so you can’t do this multiplication.
This means that anything you write will be absolute nonsense because it
is impossible to multiply these matrices in this order. This shows something you must always remember about matrix multiplication.
Order Matters!
Matrix Multiplication Is Not Commutative!
This is very different than multiplication of numbers!
Definition Let
n × p matrix. Then AB is an m × p matrix and
Example
æ
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=
ç
-
è
be an m× n matrix and let
()
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=
и
BA
12
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be an
n
=
å
1
k
=
21
æ
æ
ç
=×
BA
ç
è
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ç
÷
ç
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-
31
è
ø
12
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53
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-
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10
-
-
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æ
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=×
AB
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æ
æ
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=
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31
è
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31)2(21112
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=
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×+×-×+×
3523)1(513
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ø
АВ ≠ ВА
In this subsection, we collect properties of matrix multiplication and
its interaction with the zero matrix (Definition ZM), the identity matrix
(Definition IM), matrix addition (Definition MA), scalar matrix multiplication (Definition MSM), and the transpose (Definition TM).
Theorem MMDAA: Matrix Multiplication Distributes Across
Addition. Suppose A is an m×n matrix and B and C are n×p matrices
and D is a p×s matrix. Then
1. А(В+С)=АВ+АС
2. (В+С)D=BD+CD
Theorem MMSMM: Matrix Multiplication and Scalar Matrix
Multiplication. Suppose A is an m×n matrix and B is an
n×p matrix. Let α be a scalar. Then α(AB)=(αA)B=A(αB).
Theorem MMA: Matrix Multiplication is Associative. Suppose A
is an
m×n matrix, B is an n×p matrix and D is a p×s matrix. Then
A(BD)=(AB)D.
Theorem MMT: Matrix Multiplication and Transposes. Suppose
A is an
m×n matrix and B is an n×p matrix. Then (AB)t=BtAt.
Example
2
321
æ
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ç
ç
è
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ç
÷
-
3
311
ç
÷
ç
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542
è
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123
ö
÷
311
÷
÷
522
ø
Solution.
321
æ
ö
ç
÷
=
311
ç
÷
ç
÷
542
è
ø
ö
÷
.
÷
÷
ø
=
321
æ
æ
ç
ç
ç
è
ö
ç
÷
*
311
ç
÷
ç
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542
è
ø
æ
ç
ç
ç
è
24169
21158
432816
++++++
5*33*23*14*31*22*12*31*21*1
ö
÷
++++++
5*33*13*14*31*12*12*31*11*1
÷
÷
++++++
5*53*43*24*51*42*22*51*41*2
ø
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