Chapter 17. Vector Spaces

A vector space is a set whose elements can be added together and multiplied by scalars. The scalars usually come from $\mathbb{R}$ or $\mathbb{C}$ , but the same definition works over any field. The two operations must satisfy the vector space axioms: closure, associativity, commutativity of addition, a zero vector, additive inverses, scalar identity, scalar associativity, and distributivity.

The purpose of the definition is to isolate the algebraic behavior shared by many different objects. Arrows in the plane, columns of numbers, polynomials, matrices, and many spaces of functions can all be vector spaces. Once the axioms are known to hold, the same theorems apply to all of them.

17.1 The Definition

Let $F$ be a field. A vector space over $F$ is a set $V$ together with two operations:

u + v \in V

for vectors $u, v \in V$ , and

cv \in V

for a scalar $c \in F$ and a vector $v \in V$ .

The elements of $V$ are called vectors. The elements of $F$ are called scalars.

The word vector no longer means only an arrow or a column of numbers. It means any element of a set that satisfies the vector space rules.

17.2 Axioms

For all $u, v, w \in V$ and all $a, b \in F$ , the following properties must hold.

Axiom	Statement
Closure under addition	$u + v \in V$
Closure under scalar multiplication	$av \in V$
Associativity of addition	$(u + v) + w = u + (v + w)$
Commutativity of addition	$u + v = v + u$
Additive identity	There exists $0 \in V$ such that $v + 0 = v$
Additive inverse	For each $v \in V$ , there exists $-v \in V$ such that $v + (-v) = 0$
Scalar identity	$1v = v$
Compatibility of scalar multiplication	$a(bv) = (ab)v$
Distributivity over vector addition	$a(u + v) = au + av$
Distributivity over scalar addition	$(a + b)v = av + bv$

These axioms are not arbitrary. They say that addition behaves like ordinary addition and that scalar multiplication interacts correctly with addition.

17.3 First Examples

The space $\mathbb{R}^n$ is the standard example. Its elements are columns

v = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix},

where each $v_i$ is a real number.

Addition and scalar multiplication are defined component by component:

\begin{bmatrix} u_1 \\ \vdots \\ u_n \end{bmatrix} + \begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix} = \begin{bmatrix} u_1 + v_1 \\ \vdots \\ u_n + v_n \end{bmatrix},

and

c \begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix} = \begin{bmatrix} cv_1 \\ \vdots \\ cv_n \end{bmatrix}.

The zero vector is

0 = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}.

The additive inverse of $v$ is

-v = \begin{bmatrix} -v_1 \\ -v_2 \\ \vdots \\ -v_n \end{bmatrix}.

Thus $\mathbb{R}^n$ is a vector space over $\mathbb{R}$ .

17.4 Polynomials as Vectors

Let $P_n$ be the set of all real polynomials of degree at most $n$ :

p(x) = a_0 + a_1x + \cdots + a_nx^n.

This set is a vector space over $\mathbb{R}$ . The vectors are polynomials. The scalars are real numbers.

p(x) = 1 + 2x

and

q(x) = 3 - x,

then

p(x) + q(x) = 4 + x.

If $c = 5$ , then

cp(x) = 5 + 10x.

The zero vector is the zero polynomial:

0(x) = 0.

This example shows why the abstract definition is useful. A polynomial does not look like an arrow, but it behaves like a vector under addition and scalar multiplication.

17.5 Matrices as Vectors

Let $M_{m \times n}(\mathbb{R})$ be the set of all $m \times n$ real matrices. This set is a vector space over $\mathbb{R}$ .

Matrix addition and scalar multiplication are defined entry by entry. If

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \qquad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix},

then

A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}.

If $c = 2$ , then

2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}.

Here the vectors are matrices. The matrix shape must be fixed. The set of all $2 \times 2$ matrices forms a vector space. The union of $2 \times 2$ and $3 \times 3$ matrices does not, because addition between different shapes is undefined.

17.6 Function Spaces

Let $F(\mathbb{R}, \mathbb{R})$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ . This is a vector space over $\mathbb{R}$ .

For functions $f$ and $g$ , define

(f + g)(x) = f(x) + g(x),

and

(cf)(x) = c f(x).

The zero vector is the zero function:

0(x) = 0

for every $x$ .

For example, if

f(x) = x^2

and

g(x) = \sin x,

then

(f + g)(x) = x^2 + \sin x.

This space is usually infinite-dimensional. It contains far more vectors than $\mathbb{R}^n$ . Still, it satisfies the same vector space axioms.

17.7 Non-Examples

A set can fail to be a vector space even when it looks similar to one.

The set

S = \{(x,y) \in \mathbb{R}^2 : x \geq 0\}

is not a vector space over $\mathbb{R}$ . It is closed under addition, but not under scalar multiplication. For example,

(1,0) \in S,

but

-1(1,0) = (-1,0) \notin S.

The set

T = \{(x,y) \in \mathbb{R}^2 : x + y = 1\}

is not a vector space. It does not contain the zero vector, since

0 + 0 \neq 1.

It is an affine line, not a vector space.

The set of polynomials of exact degree $2$ is not a vector space. For example,

x^2 + (-x^2) = 0,

and the zero polynomial does not have exact degree $2$ .

These examples show that closure and the zero vector are often the fastest tests.

17.8 Linear Combinations

A linear combination of vectors $v_1, v_2, \ldots, v_k$ is a vector of the form

c_1v_1 + c_2v_2 + \cdots + c_kv_k,

where $c_1, c_2, \ldots, c_k$ are scalars.

Linear combinations are the basic expressions of linear algebra. They are built using only the two vector space operations: addition and scalar multiplication.

For example, in $\mathbb{R}^3$ ,

2 \begin{bmatrix} 1 \\ 0 \\ 3 \end{bmatrix} - \begin{bmatrix} 4 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ 6 \end{bmatrix} - \begin{bmatrix} 4 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} -2 \\ -1 \\ 4 \end{bmatrix}.

The result is again a vector in $\mathbb{R}^3$ .

17.9 Span

The span of a set of vectors is the set of all linear combinations of those vectors.

If $v_1, v_2, \ldots, v_k \in V$ , then

\operatorname{span}(v_1,\ldots,v_k) = \{c_1v_1 + \cdots + c_kv_k : c_i \in F\}.

The span is always a vector space inside $V$ . More precisely, it is a subspace of $V$ .

For example, in $\mathbb{R}^2$ , let

v = \begin{bmatrix} 1 \\ 2 \end{bmatrix}.

Then

$$
\operatorname{span}(v) =
\left{
c
\begin{bmatrix}
1 \
2
\end{bmatrix}: c \in \mathbb{R} \right}. $$

This is the line through the origin in the direction of $v$ .

17.10 Subspaces

A subspace of a vector space $V$ is a subset $W \subseteq V$ that is itself a vector space under the same operations.

To prove that $W$ is a subspace, it is enough to check three conditions:

Condition	Meaning
Nonempty	$0 \in W$
Closed under addition	If $u, v \in W$ , then $u + v \in W$
Closed under scalar multiplication	If $v \in W$ and $c \in F$ , then $cv \in W$

For example,

W = \{(x,y,z) \in \mathbb{R}^3 : x + y + z = 0\}

is a subspace of $\mathbb{R}^3$ .

It contains the zero vector because

0 + 0 + 0 = 0.

If $u$ and $v$ satisfy the equation, then so does $u + v$ . If $u$ satisfies the equation, then so does $cu$ . Therefore $W$ is a subspace.

17.11 Linear Independence

Vectors $v_1, v_2, \ldots, v_k$ are linearly independent if the equation

c_1v_1 + c_2v_2 + \cdots + c_kv_k = 0

has only the trivial solution

c_1 = c_2 = \cdots = c_k = 0.

They are linearly dependent if there is a nontrivial solution. That means at least one coefficient is not zero.

Linear independence means that no vector in the list is redundant. Each vector contributes a new direction that cannot be built from the others.

For example,

\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad \begin{bmatrix} 0 \\ 1 \end{bmatrix}

are linearly independent in $\mathbb{R}^2$ .

But

\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad \begin{bmatrix} 2 \\ 0 \end{bmatrix}

are linearly dependent, since

2 \begin{bmatrix} 1 \\ 0 \end{bmatrix} - \begin{bmatrix} 2 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}.

17.12 Basis

A basis of a vector space is a list of vectors that is both spanning and linearly independent.

The list spans the space, so every vector can be built from it. The list is linearly independent, so no vector in the list is redundant.

In $\mathbb{R}^3$ , the standard basis is

e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \qquad e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \qquad e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.

Every vector

v = \begin{bmatrix} a \\ b \\ c \end{bmatrix}

can be written uniquely as

v = ae_1 + be_2 + ce_3.

This uniqueness is the main reason bases are important. A basis turns abstract vectors into coordinates.

17.13 Dimension

The dimension of a vector space is the number of vectors in any basis.

For example,

\dim(\mathbb{R}^n) = n.

The vector space $P_n$ of polynomials of degree at most $n$ has dimension $n+1$ , because a basis is

1, x, x^2, \ldots, x^n.

The space $M_{m \times n}(\mathbb{R})$ has dimension $mn$ , because each matrix has $mn$ independent entries.

Dimension measures the number of independent parameters needed to describe a vector in the space.

17.14 Coordinates

Once a basis is chosen, every vector has coordinates.

Let

B = (v_1, v_2, \ldots, v_n)

be a basis of $V$ . If

v = c_1v_1 + c_2v_2 + \cdots + c_nv_n,

then the coordinate vector of $v$ with respect to $B$ is

[v]_B = \begin{bmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{bmatrix}.

Coordinates depend on the basis. The vector itself does not.

This distinction is important. A vector is an object in a vector space. A coordinate vector is a representation of that object after choosing a basis.

17.15 The Zero Vector

Every vector space has exactly one zero vector.

The zero vector satisfies

v + 0 = v

for every $v \in V$ .

Although the symbol $0$ is used in many spaces, its meaning depends on the space.

Space	Zero vector
$\mathbb{R}^n$	$(0,\ldots,0)$
$P_n$	Zero polynomial
$M_{m \times n}$	Zero matrix
Function space	Zero function

The zero vector is required for a vector space. Any candidate set that lacks it cannot be a vector space.

17.16 Additive Inverses

Every vector $v$ in a vector space has an additive inverse $-v$ , satisfying

v + (-v) = 0.

In $\mathbb{R}^n$ , this means changing the sign of every component. For polynomials, it means changing the sign of every coefficient. For functions, it means defining

(-f)(x) = -f(x).

Additive inverses allow subtraction to be defined:

u - v = u + (-v).

Thus subtraction is not a primitive operation in the definition. It is derived from addition and additive inverse.

17.17 Consequences of the Axioms

Several useful facts follow from the axioms.

For every scalar $c$ and vector $v$ ,

0v = 0,

c0 = 0,

(-1)v = -v.

Also, if

cv = 0,

then either

c = 0

v = 0.

These facts are not separate axioms. They are theorems proved from the vector space axioms.

As an example, prove that $0v = 0$ . Since $0 + 0 = 0$ in the scalar field,

0v = (0 + 0)v.

By distributivity,

(0 + 0)v = 0v + 0v.

Thus

0v = 0v + 0v.

Add the additive inverse of $0v$ to both sides. The result is

0 = 0v.

Therefore

0v = 0.

This proof shows how the axioms control all ordinary algebraic behavior.

17.18 Vector Spaces over Different Fields

The field of scalars matters.

The space $\mathbb{R}^n$ is naturally a vector space over $\mathbb{R}$ . The space $\mathbb{C}^n$ is naturally a vector space over $\mathbb{C}$ . It can also be viewed as a vector space over $\mathbb{R}$ , but then its dimension changes.

For example, $\mathbb{C}$ has dimension $1$ over $\mathbb{C}$ , with basis

1.

But $\mathbb{C}$ has dimension $2$ over $\mathbb{R}$ , with basis

1, i.

Every complex number can be written as

a + bi.

Here $a$ and $b$ are real coordinates.

Thus the same set can define different vector spaces depending on the field of scalars.

17.19 Finite and Infinite Dimensional Spaces

A vector space is finite-dimensional if it has a finite basis. It is infinite-dimensional if no finite basis spans it.

The spaces

\mathbb{R}^n, \qquad P_n, \qquad M_{m \times n}(\mathbb{R})

are finite-dimensional.

The space of all real polynomials is infinite-dimensional. A finite list of polynomials has a maximum degree. It cannot span polynomials of higher degree.

The space of all functions from $\mathbb{R}$ to $\mathbb{R}$ is also infinite-dimensional.

Finite-dimensional spaces are the main subject of elementary linear algebra. Infinite-dimensional spaces appear naturally in analysis, differential equations, Fourier theory, and functional analysis.

17.20 Summary

A vector space is a set with addition and scalar multiplication satisfying precise algebraic laws. The definition includes many objects that at first look different: coordinate vectors, matrices, polynomials, and functions.

The main ideas introduced in this chapter are:

Concept	Meaning
Vector space	A set closed under vector addition and scalar multiplication
Scalar	An element of the underlying field
Linear combination	A sum of scalar multiples of vectors
Span	The set of all linear combinations
Subspace	A subset that is itself a vector space
Linear independence	No vector is redundant
Basis	A linearly independent spanning list
Dimension	The number of vectors in a basis
Coordinates	Scalars describing a vector relative to a basis

Vector spaces provide the language for the rest of linear algebra. Once this structure is fixed, matrices become representations of linear maps, systems of equations become questions about subspaces, and dimension becomes the measure of independent direction.