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:

Commutative Property of Addition:
Associative Property of Addition:
Existence of Additive Identity:
Existence of Additive Inverse:
Distributive Property over Vector Addition:
Distributive Property over Scalar Addition:
Associative Property for Scalar Multiplication:
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.

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.

Sets of polynomials provide an important source of examples, so we review some basic facts. A 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,

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.


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.

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 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

See Practice Problem prob:spanisasubspaceabstract.

Practice Problems

(Adapted from Kuttler, Exercises 9.1.1-9.1.4) 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.
Scalar Multiplication:
Scalar Multiplication:
Scalar Multiplication:
Scalar Multiplication:
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:

A solution to such an equation is a function.

Verify that is a solution to (diffeq).
Is a solution?
Is a solution?
Is a solution?
Let be the set of all solutions to (diffeq). Prove that is a vector space.
In this problem we will check that the set of all complex numbers is in fact a vector space. Let be a complex number where . Similarly, let and 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.
Refer to Example ex:centralizerofA and describe all elements of , where is a identity matrix.
Is the subset of all invertible matrices a subspace of ? Prove your claim.
Is the subset of all symmetric (See Definition def:symmetricandskewsymmetric of MAT-0025) matrices a subspace of ? 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 ?

Don’t forget to check that is not empty!
List several elements of . Suggest a spanning set for .
Find .

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.