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 :
In the next two examples we will explore two sets other than endowed with addition and scalar multiplication and satisfying the same properties.
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 :
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
- (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
- (f)
- Distributive Property over Scalar Addition:
This property holds because
- (g)
- Associative Property for Scalar Multiplication:
This property holds because
- (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.
- 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 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
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,
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.
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.
Subspaces
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.
- 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 .
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 .
Linear Combinations and Span
Turning to , we are looking for and such that
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
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.