Chapter 4: Vector Spaces – Part I
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Chapter 4: Vector Spaces – Part I
The two most important operations in mathematics, in particular linear
algebra, are addition and scalar multiplication. Both operations follow the
same set of arithmetic rules. It is worth to develop a general theory of
mathematical system for addition and scalar multiplication. Vector (Linear)
spaces are the mathematical systems that can serve the purpose. This chapter
will present the formal definition of a vector space and discuss some of the
concepts related to vector spaces, for instances, subspaces. Examples and
exercises are included. Topics included in this chapter are listed as follows:
Section 1.1: Definition of Vector Spaces
Section 1.2: Subspaces
Section 1.1: Definition of Vector Spaces
1. Question: What is a vector space?
Answer: A vector space is a mathematical system for addition
and scalar multiplication.
2. Formal definition of vector spaces:
Suppose V is a non-empty set. Then, we define on V the operations of
addition and scalar multiplication.
That is, for any two elements x and y in V, there is a unique element x + y
in V; For any number a, there is a unique element ax in V.
Definition: The set V equipped with addition and scalar multiplication is
said to be a vector space if it satisfies the following eight axioms:
Chapter 4: Vector Spaces – Part I
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(Axioms of addition)

x, y  V
(A1)
x + y = y + x,
(A2)
x + (y + z) = (x + y) + z,
(A3)

(A4)


x, y, z  V
0  V such that x + 0 = x
x  V,

-x  V such that x + (-x) = 0
(Axioms of multiplication)
(M1)
a (x + y) = ax + ay,

x, y  V,
(M2)
(a + b) x = ax + bx,

x  V,
(M3)
a (b x) = (a b) x,
(M4)
1 x = x,


x  V,



aR
a, b  R
a, b  R
xV
We call each element in a vector space a vector
3. Examples of vector spaces:
 Euclidean space Rn
Every element in Rn is an n-dimensional vector with each entity
being a real number
Recall that for x := (x1, x2,
, xn) and y := (y1, y2,
, yn) in
Rn, the operation of addition is defined as:
(x1, x2,
, xn) + (y1, y2,
, yn) = (x1 + y1, x2 + y2,
, xn + yn)
Chapter 4: Vector Spaces – Part I
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For any scalar a  R,
a(x1, x2,
, xn) = (ax1, ax2,
, axn)
We can also define the zero vector
0 = (0, 0,
, 0)
and the negative vector
-(x1, x2,
, xn) = (-x1, -x2,
, -xn)
 Exercise: Check whether the Euclidean space Rn satisfies the eight
axioms of a vector space.
 The space of all polynomials P(x)
Suppose P(x) denote the space of all polynomials of the following
form:
p(x) = a0 + a1 x + a2 x2 + …. + an xn, n  N
The addition of two polynomials
p(x) = a0 + a1 x + a2 x2 + …. + an xn
and
q(x) = b0 + b1 x + b2 x2 + …. + bn xn
can be defined as follows:
p(x)+ q(x)
= (a0 + b0) + (a1 + b1) x + ( a2 + b2) x2 + …. + ( an + bn) xn
Chapter 4: Vector Spaces – Part I
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The scalar multiplication of a polynomial
Let r be a scalar (i.e. r  R)
Then, the multiplication of a polynomial p(x) by a scalar r is defined
as follows:
r p(x) = (a0 r)+(a1r) x + (a2r) x2 + …. + (anr)xn
We can define the zero vector 0 by the zero polynomial (i.e. the
polynomial with all coefficients being zero)
The negative vector of the polynomial p(x) is defined by:
-p(x) = - a0 - a1 x - a2 x2 + …. - an xn
It is easy to check that
p(x) + (-p(x)) = 0
 The space Pn(x) of all polynomials with degree less than or equal to
n
An element p(x) of Pn(x) can be expressed in the following form:
p(x) = a0 + a1 x + a2 x2 + …. + an xs, s

n, s  N
The operations of addition and scalar multiplication can be defined
in a usual way.
The zero vector is also the zero polynomial
 Exercise: Check whether the space Pn(x) satisfies the eight axioms
of a vector space.
Chapter 4: Vector Spaces – Part I
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 The space M(m, n) of all m × n matrices:
The addition and the scalar multiplication of matrices in M(m, n) are
defined by the usual addition and scalar multiplication of matrices
The zero vector in M(m, n) are the m × n zero matrix
 Exercise: Check whether the space M(m, n) satisfies the eight
axioms of a vector space.
 The space F(D) of all real-valued functions f: D  R
The addition of two functions f and g in F(D):
(f+g) (x) = f(x) + g(x),

x
D
The multiplication of a function f in F(D) by a scalar a
(af)(x) = af(x),

x
R:
D
The zero vector f in F(D) is defined by the zero function:
f(x) = 0,

x
D
The negative function of a given function f in F(D):
(-f)(x) = -f(x) x D
 Exercise: Check whether the space F(D) satisfies the eight axioms of
a vector space.
4. Counterexamples of vector spaces:
 The space of P5(x) of all polynomials of degree 5
For each p(x) in P5(x), we have
p(x) = a0 + a1 x + a2 x2 + a3 x3 + a4x4 + a5 x5
Chapter 4: Vector Spaces – Part I
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Question: Which axioms of a vector space are not satisfied
by P5(x)?
Answer: (A3) is not satisfied by P5(x) since it does not the zero
element (i.e. The degree of a zero polynomial is zero, but all
polynomials in P5(x) are of degree five)
Besides (A3), the operation of addition cannot always be defined in
P5(x). For instance,
Consider two following polynomials f and g in P5(x):
f(x) = x4 – 2x5
and
g(x) = 2x5
The addition of f and g is given by:
(f + g)(x) = x4
It is not a polynomial with degree 5
5. Some important properties of vector spaces
 Suppose V is a vector space
 x,
y, z  V,
x + z = y + z => x = y

a 
R,
a0=0
Chapter 4: Vector Spaces – Part I
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Proof:
a 0 = a (0+0) = a0 + a0
=> a 0 = 0

(by property (1))
 x  V,
0x=0
Proof:
0x = (0 + 0)x = 0x + 0x
=> 0 = 0x
(by Property (1))
 Suppose a 0 and ax = 0, where x V
Then,
x=0
Proof:
x = 1x = (a-1a) x = a-1(a x) = a-10 = 0

a
R and x V, (-a)x = a(-x)
Proof:
0 = a0 = a(x + (-x)) = au + a(-x)
and
0 = 0x = (a + (-a))x = ax + (-a)x
Hence,
a(-x) = (-a)x
Chapter 4: Vector Spaces – Part I
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Section 1.2: Subspaces
1. Suppose S is a non-empty subset of a vector space V and it satisfies the
following conditions:
  a  R, ax  S
  x, y  S, x + y  S
 0S
Then, we call S a subspace of V
2. The interpretation of the first condition is that S is closed under scalar
multiplication
The interpretation of the second condition is that S is closed under
addition
Hence, we can interpret the subspace S as a subset of V that is closed
under the operations of addition and scalar multiplication of V
3. Examples and Counterexamples of Subspaces:
 Consider the two-dimensional Euclidean space
Note that
2
2
:
is a vector space
Let S1 = {(x, 0) | x
} (i.e. The set of ordered pairs with the
second entity being zero)
Exercise: Check that S1 is a subspace of
2
Chapter 4: Vector Spaces – Part I
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 Consider again a two-dimensional Euclidean space
Let S2 = {(2, y) | y
element being 2)
2
} (i.e. The set of ordered pairs with the first
Exercise: Is S2 a subspace of
2
? Give reason(s).
 Consider the following subset S2 of
2
:
Let S3 = {(x, y)|x, y , x = y} (i.e. The set of ordered pairs with the
first element being equal to the second element)
Exercise: Is S3 a subspace of
 Any straight in
2
2
? Give reason(s).
passing through the origin (0, 0) is a subspace
Exercise: Check the above statement.
 Any plane in
3
containing the origin (0, 0, 0) is also a subspace
Exercise: Check the above statement.
 Consider the following subset of S4 of
2
:
Let S4 = {(y, y2) | y
} (i.e. The set of ordered pairs with the
second entity being the square of the first entity
Exercise: Is S4 a subspace of
2
? Give reason(s).