Addition:
Scalar Multiplication:
Abstract Vector Spaces
In Subspaces of we discussed as a vector space and introduced the notion of a subspace of . In this section we will consider sets other than that have two operations and satisfy the same properties as . Such sets, together with the operations of addition and scalar multiplication, will also be called vector spaces.
Properties of Vector Spaces
Recall that is said to be a vector space because
- is closed under vector addition
- is closed under scalar multiplication
and satisfies the following properties:
- (a)
- Commutative Property of Addition:
- (b)
- Associative Property of Addition:
- (c)
- Existence of Additive Identity:
- (d)
- Existence of Additive Inverse:
- (e)
- Distributive Property over Vector Addition:
- (f)
- Distributive Property over Scalar Addition:
- (g)
- Associative Property for Scalar Multiplication:
- (h)
- Multiplication by :
All scalars in this chapter are assumed to be real numbers. Complex scalars are
considered in Complex Matrices.
In the next two examples we will explore two sets other than endowed with addition and scalar multiplication and satisfying the same properties.
Let be the set of all matrices. Matrix addition and scalar multiplication were
defined in Addition and Scalar Multiplication of Matrices.
Observe that the sum of two matrices is also an matrix. Likewise, a scalar multiple of an matrix is an matrix. Thus
- is closed under matrix addition
- is closed under scalar multiplication
In addition, Theorems th:propertiesofaddition and th:propertiesscalarmult give us the following properties of matrix addition and scalar multiplication. Note that these properties are analogous to the eight vector properties above.
- (a)
- Commutative Property of Addition:
- (b)
- Associative Property of Addition:
- (c)
- Existence of Additive Identity: where is the zero matrix
- (d)
- Existence of Additive Inverse:
- (e)
- Distributive Property over Matrix Addition:
- (f)
- Distributive Property over Scalar Addition:
- (g)
- Associative Property for Scalar Multiplication:
- (h)
- Multiplication by :
Consider the set of all linear functions. This set includes all polynomials of degree
and degree . We will use addition and scalar multiplication of polynomials as the two
operations, and show that is closed under those operations and satisfies eight
properties analogous to those of vectors of .
Elements of are functions given
by
(Note that and can be equal to zero.)
Given and in , it is easy to verify that is also in . This gives us closure under function addition.
For any scalar , we have Therefore is in , and is closed under scalar multiplication.
We now proceed to formulate eight properties analogous to those of vectors of .
Let , and be elements of given by , , and . Let and be scalars.
- (a)
- Commutative Property of Addition:
This property holds because \begin{align*} f_1(x) + f_2(x) &= (m_1 x + b_1) + (m_2 x + b_2)\\ &= (m_2 x + b_2) + (m_1 x + b_1)\\ &= f_2(x) + f_1(x) \end{align*}
- (b)
- Associative Property of Addition:
This property is easy to verify.
- (c)
- Existence of Additive Identity:
The additive identity is given by . Note that is in .
- (d)
- Existence of Additive Inverse:
The additive inverse of is a function given by . Note that is in .
- (e)
- Distributive Property over Vector Addition:
This property holds because \begin{align*} k(f_1(x) + f_2(x)) &= k((m_1 x + b_1) + (m_2 x + b_2))\\ &= k(m_1 x + b_1) + k(m_2 x + b_2)\\ &= k f_1(x) + k f_2(x) \end{align*}
- (f)
- Distributive Property over Scalar Addition:
This property holds because \begin{align*} (k+p)f_1(x)&= (k+p)(m_1 x + b_1)\\ &=k(m_1 x + b_1) + p(m_1 x + b_1)\\ &= k f_1(x) + p f_1(x) \end{align*}
- (g)
- Associative Property for Scalar Multiplication:
This property holds because \begin{align*} k(p(f_1(x)))&=k(p(m_1 x + b_1))\\ &=k(p m_1 x +p b_1)\\ &=kp m_1 x +kp b_1\\ &= (kp) m_1 x + (kp) b_1\\ &= (kp)(m_1 x + b_1)\\ &=(kp)f_1(x) \end{align*}
- (h)
- Multiplication by
Definition of a Vector Space
Examples ex:setofmatricesvectorspace and ex:linfunctionsvectspace show us that there are many times in mathematics when we encounter a set with two operations (that we call addition and scalar multiplication) such that the set is closed under the two operations, and satisfies the same eight properties as . We will refer to such sets as vector spaces.
Let be a nonempty set. Suppose that elements of can be added together and
multiplied by scalars. The set , together with operations of addition and scalar
multiplication, is called a vector space provided that
- is closed under addition
- is closed under scalar multiplication
and the following properties hold for , and in and scalars and :
- (a)
- Commutative Property of Addition:
- (b)
- Associative Property of Addition:
- (c)
- Existence of Additive Identity:
- (d)
- Existence of Additive Inverse:
- (e)
- Distributive Property over Vector Addition:
- (f)
- Distributive Property over Scalar Addition:
- (g)
- Associative Property for Scalar Multiplication:
- (h)
- Multiplication by :
We will refer to elements of as vectors.
When scalars and in the above definition are restricted to real numbers, as they are in this chapter, vector space may be referred to as a vector space over the real numbers.
Sets of polynomials provide an important source of examples, so we review some basic facts. A polynomial with real coefficients in is an expression \begin{equation*} p(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n \end{equation*} where are real numbers called the coefficients of the polynomial. If all the coefficients are zero, the polynomial is called the zero polynomial and is denoted simply as . If , the highest power of with a nonzero coefficient is called the degree of denoted as . The coefficient itself is called the leading coefficient of . Hence , , and . (The degree of the zero polynomial is not defined.)Let denote the set of all polynomials and suppose that \begin{align*} p(x) &= a_0 + a_1x + a_2x^2 + \ldots \\ q(x) &= b_0 + b_1x + b_2x^2 + \ldots \end{align*}
are two polynomials in (possibly of different degrees). Then and are called equal (written ) if and only if all the corresponding coefficients are equal—that is, , , , and so on. In particular, means that , , , .
The set has an addition and scalar multiplication defined on it as follows: if and are as before and is a real number, \begin{align*} p(x) + q(x) &= (a_0 + b_0) + (a_1 + b_1)x + (a_2 + b_2)x^2 + \ldots \\ kp(x) &= ka_0 + (ka_1)x + (ka_2)x^2 + \ldots \end{align*}
Let be the set of all degree two polynomials in . In other words,
We claim that is not a vector space.
Observe that is not closed under addition. To
see this, let and let . Then and are both elements of . However, is not an
element of , as it is only a degree one polynomial. We require the coefficient
of to be nonzero for a polynomial to be in , and this is not the case for
.
As an exercise, check the remaining vector space properties one-by-one to see which properties hold and which do not.
Set in Example ex:deg2onlynotavecspace is not a vector space, but if we make a slight modification, we can make it into a vector space.
Let be the set of polynomials of degree two or less. In other words,
Note that contains the zero polynomial (let ).
Unlike set in Example ex:deg2onlynotavecspace, is closed under polynomial addition and scalar multiplication. It is easy to verify that all vector space properties hold, so is a vector space.
Let be a natural number. Define to be the set of polynomials of degree or less than
, then by reasoning similar to Example ex:deg_le_2vectorspace, is a vector space.
Subspaces
A nonempty subset of a vector space is called a subspace of , provided that is itself
a vector space when given the same addition and scalar multiplication as
.
In Example ex:deg_le_2vectorspace we demonstrated that is a vector space. From Example ex:pisavectorspace we know that
is a vector space. But is a subset of , and uses the same operations of
polynomial addition and scalar multiplication. Therefore is a subspace of
.
Checking all ten properties to verify that a subset of a vector space is a subspace can be cumbersome. Fortunately we have the following theorem.
Subspace Test Let be a nonempty subset of a vector space . If is closed under the
operations of addition and scalar multiplication of , then is a subspace of
.
- Proof
- To prove that closure is a sufficient condition for to be a subspace,
we will need to show that closure under addition and scalar multiplication of
guarantees that the remaining eight properties are satisfied automatically.
Observe that Properties item:commaddvectspdef, item:assaddvectspdef, item:distvectaddvectspdef, item:distscalaraddvectspdef, item:assmultvectspdef and item:idmultvectspdef hold for all elements of . Thus, these properties will hold for all elements of . We say that these properties are inherited from .
To prove Property item:idaddvectspdef we need to show that , which we know to be an element of , is contained in . Let be an element of (recall that is nonempty). We will show that in . Then, by closure under scalar multiplication, we will be able to conclude that must be in . Adding the additive inverse of to both sides gives us By Properties item:assaddvectspdef and item:invaddvectspdef
By Properties item:idaddvectspdef and item:invaddvectspdef
Because is closed under scalar multiplication is in .
We know that every element of , being an element of , has an additive inverse in . We need to show that the additive inverse of every element of is contained in . Let be any element of . We will show that is the additive inverse of . Then by closure, will have to be contained in . To show that is the additive inverse of , we must show that . We compute: Thus is the additive inverse of . By closure, is in .
Let be a fixed matrix in . Show that the set of all matrices that commute with
under matrix multiplication is a subspace of .
The set consists of all matrices such that . First, observe that is not empty because
is an element. Now we need to show that is closed under matrix addition and scalar
multiplication.
Suppose that and lie in . Then and . Then Therefore commutes with . Thus is in . We conclude that is closed under matrix addition.
Now suppose is in . Let be a scalar, then Therefore commutes with . We conclude that is in , and is closed under scalar multiplication. Hence is a subspace of .
Suppose is a polynomial and is a number. Then the number obtained by replacing by in the expression for is called the evaluation of at . For example, if , then the evaluation of at is . If , the number is called a root of .
Consider the set of all polynomials in that have as a root: \begin{equation*} U = \{p(x) \in \mathbb{P} : p(3) = 0 \} \end{equation*}
Show that is a subspace of .
Linear Combinations and Span
Let be a vector space and let be vectors in . A vector is said to be a linear
combination of vectors if
for some scalars .
Let be a vector space and let be vectors in . The set of all linear combinations of
is called the span of . We write
and we say that vectors span . Any vector in is said to be in the span of . The set
is called a spanning set for .
Consider and in . Determine whether and lie in .
For , we want to determine if and exist such that \begin{equation*} p_1 = a(1 + 2x - x^2) + b(3 + 5x + 2x^2) \end{equation*}
Expanding the right hand side gives us:
Combining like terms, we get:
Setting this equal to and equating coefficients of powers of gives us a system of
equations \begin{equation*} 1 = a + 3b,\quad 1 = 2a + 5b, \quad \mbox{ and } \quad 4 = -a + 2b \end{equation*}
This system has the solution and , so is indeed in .
Turning to , we are looking for and such that \begin{equation*} p_{2} = a(1 + 2x - x^{2}) + b(3 + 5x + 2x^{2}) \end{equation*} Again equating coefficients of powers of gives equations , , and . But in this case there is no solution, so is not in .
- Proof
- See Practice Problem prob:spanisasubspaceabstract.
Practice Problems
Problems prob:abstractvectspace1-prob:abstractvectspace4
Is the set of all points in a vector space under the given definitions of addition and scalar multiplication? In each case be specific about which vector space properties hold and which properties fail.
Let be the set of all real-valued functions whose domain is all real numbers. Define
addition and scalar multiplication as follows:
Verify that is a vector space.
A differential equation is an equation that contains derivatives. Consider the
differential equation: \begin{align} \label{diffeq} f''+f=0 \end{align}
A solution to such an equation is a function.
In this problem we will check that the set of all complex numbers is in fact a vector
space. Let be a complex number. Similarly, let , be complex numbers, and let
and be real number scalars. Check that complex numbers are closed under
addition and multiplication, and that they satisfy each of the vector space
properties.
Is the subset of all symmetric matrices a subspace of ? (See Definition def:symmetricandskewsymmetric.) Prove your
claim.
Let be a subset of that consists of matrices that commute with every matrix in
under matrix multiplication. In other words,
Is a subspace of ?
Text Source
The discussion on polynomials was adapted from Section 6.1 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p. 331-332
Example Source
Examples ex:root3 and ex:inthespanpoly were adapted from Examples 6.2.4 and 6.2.7 of Keith Nicholson’s Linear Algebra with Applications. (CC-BY-NC-SA)
W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2018, Open Edition, p. 340-341
Exercise Source
Practice Problems prob:abstractvectspace1-prob:abstractvectspace4 is adopted from Problems 9.1.1-9.1.4 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)
Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, p. 469.
2024-09-07 16:13:07