We state the definition of an abstract vector space, and learn how to determine if a given set with two operations is a vector space. We define a subspace of a vector space and state the subspace test. We find linear combinations and span of elements of a vector space.

### VSP-0050: Abstract Vector Spaces

#### Properties of Vector Spaces

In VSP-0020 we discussed as a vector space and introduced the notion of a subspace of . In this module we will consider sets other than that have two operations and satisfy the same properties. Such sets, together with the operations of addition and scalar multiplication, will also be called vector spaces.

Recall that was 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 of MAT-0010 give us the following properties of matrix addition and scalar multiplication. Note that these properties are analogous to the eight vector properties.

- (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)
that satisfies the same properties as . We will refer to such sets as *vector
spaces*.

*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*.

*polynomial*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

*subspace*of , provided that is itself a vector space when given the same addition and scalar multiplication as .

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

*linear combination*of vectors if for some scalars .

*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 .

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

A solution to such an equation is a function.

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.