# Chapter 2. Scalars, Vectors, and Fields

# Chapter 2. Scalars, Vectors, and Fields

Linear algebra uses two kinds of objects: scalars and vectors.

Scalars are the numbers used for measurement and scaling. Vectors are the objects being added and scaled. The word field names the number system from which the scalars are taken. A vector space is therefore always a vector space over some field. The usual examples are real vector spaces over \(\mathbb{R}\) and complex vector spaces over \(\mathbb{C}\). More generally, the scalars may come from any field.

## 2.1 Scalars

A scalar is a single element of the field of scalars.

In elementary linear algebra, the scalars are usually real numbers. Thus \(2\), \(-5\), \(0\), \(\frac{3}{4}\), and \(\pi\) are scalars. In complex linear algebra, numbers such as

$$
3 + 2i
$$

are also scalars.

Scalars multiply vectors. If \(c\) is a scalar and \(v\) is a vector, then \(cv\) is again a vector.

For example,

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

The scalar changes the size of the vector. A negative real scalar also reverses its direction in the geometric interpretation.

## 2.2 Vectors

A vector is an element of a vector space.

In \(\mathbb{R}^n\), a vector is an ordered list of \(n\) real numbers:

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

The entries \(v_1,\ldots,v_n\) are called the components of \(v\).

For example,

$$
\begin{bmatrix}
4 \\
0 \\
-2
\end{bmatrix}
$$

is a vector in \(\mathbb{R}^3\).

A vector can represent a point, a displacement, a force, a velocity, a list of features, a row of data, or a solution to a system of equations. Linear algebra does not depend on the interpretation. It depends on the operations that vectors support.

## 2.3 Fields

A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are possible.

The most common fields are:

| Field | Description |
|---|---|
| \(\mathbb{Q}\) | Rational numbers |
| \(\mathbb{R}\) | Real numbers |
| \(\mathbb{C}\) | Complex numbers |
| \(\mathbb{F}_p\) | Integers modulo a prime \(p\) |

The field controls the scalars. When we say that \(V\) is a vector space over \(F\), we mean that vectors in \(V\) are multiplied by scalars from \(F\). The scalar system and the vector system are distinct: the field has its own addition and multiplication, while the vector space has vector addition and scalar multiplication.

## 2.4 Vector Spaces over a Field

A vector space over a field \(F\) is a set \(V\) whose elements are called vectors, together with two operations:

$$
u + v
$$

and

$$
cv,
$$

where \(u,v \in V\) and \(c \in F\).

The first operation adds two vectors. The second multiplies a vector by a scalar. These operations must satisfy the vector space axioms: associativity, commutativity of vector addition, existence of a zero vector, existence of additive inverses, distributive laws, and compatibility with scalar multiplication.

The notation

$$
V \text{ over } F
$$

should always be read as: the vectors are in \(V\), and the scalars are in \(F\).

## 2.5 The Basic Example: \(\mathbb{R}^n\)

The set \(\mathbb{R}^n\) is the set of all ordered lists of \(n\) real numbers.

If

$$
u =
\begin{bmatrix}
u_1 \\
u_2 \\
\vdots \\
u_n
\end{bmatrix},
\qquad
v =
\begin{bmatrix}
v_1 \\
v_2 \\
\vdots \\
v_n
\end{bmatrix},
$$

then vector addition is defined component by component:

$$
u+v =
\begin{bmatrix}
u_1+v_1 \\
u_2+v_2 \\
\vdots \\
u_n+v_n
\end{bmatrix}.
$$

If \(c \in \mathbb{R}\), then scalar multiplication is

$$
cv =
\begin{bmatrix}
cv_1 \\
cv_2 \\
\vdots \\
cv_n
\end{bmatrix}.
$$

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

## 2.6 The Zero Vector

Every vector space has a zero vector. It is the vector that changes nothing when added to another vector.

In \(\mathbb{R}^n\), the zero vector is

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

For every vector \(v\),

$$
v + 0 = v.
$$

The zero vector depends on the vector space. The scalar \(0\) belongs to the field. The zero vector belongs to the vector space. They are written with the same symbol when the meaning is clear, but they are different kinds of objects.

## 2.7 Additive Inverses

For every vector \(v\), there is a vector \(-v\) such that

$$
v + (-v) = 0.
$$

If

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

then

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

Subtraction is defined by addition of an inverse:

$$
u - v = u + (-v).
$$

This definition is important because vector subtraction is not a separate primitive operation. It is derived from vector addition and additive inverses.

## 2.8 Scalar Multiplication and Structure

Scalar multiplication must interact correctly with vector addition and field multiplication.

For scalars \(a,b \in F\) and vectors \(u,v \in V\), the following identities hold:

$$
a(u+v) = au + av,
$$

$$
(a+b)v = av + bv,
$$

$$
(ab)v = a(bv),
$$

$$
1v = v.
$$

These rules say that scalar multiplication is compatible with both the algebra of the field and the algebra of the vector space.

For example,

$$
2
\left(
\begin{bmatrix}
1 \\
4
\end{bmatrix}
+
\begin{bmatrix}
3 \\
-1
\end{bmatrix}
\right) =
2
\begin{bmatrix}
4 \\
3
\end{bmatrix} =
\begin{bmatrix}
8 \\
6
\end{bmatrix}.
$$

On the other hand,

$$
2
\begin{bmatrix}
1 \\
4
\end{bmatrix}
+
2
\begin{bmatrix}
3 \\
-1
\end{bmatrix} =
\begin{bmatrix}
2 \\
8
\end{bmatrix}
+
\begin{bmatrix}
6 \\
-2
\end{bmatrix} =
\begin{bmatrix}
8 \\
6
\end{bmatrix}.
$$

The same result appears on both sides.

## 2.9 Real and Complex Vector Spaces

A real vector space is a vector space whose scalars are real numbers.

A complex vector space is a vector space whose scalars are complex numbers.

The difference matters. The same set may behave differently depending on the field of scalars.

For example, \(\mathbb{C}\) is a one-dimensional vector space over \(\mathbb{C}\), because every complex number \(z\) can be written as

$$
z \cdot 1.
$$

But \(\mathbb{C}\) is a two-dimensional vector space over \(\mathbb{R}\), because every complex number can be written as

$$
a + bi = a \cdot 1 + b \cdot i,
$$

where \(a,b \in \mathbb{R}\).

Thus dimension depends on the field.

## 2.10 Finite Fields

Linear algebra can also be done over finite fields.

The simplest finite fields are \(\mathbb{F}_p\), where \(p\) is prime. The elements are

$$
0,1,2,\ldots,p-1,
$$

with arithmetic performed modulo \(p\).

For example, in \(\mathbb{F}_5\),

$$
3 + 4 = 2
$$

because \(7 \equiv 2 \pmod 5\), and

$$
3 \cdot 4 = 2
$$

because \(12 \equiv 2 \pmod 5\).

A vector over \(\mathbb{F}_5\) may look like

$$
\begin{bmatrix}
1 \\
4 \\
2
\end{bmatrix}.
$$

All vector addition and scalar multiplication are performed modulo \(5\).

Finite fields are important in coding theory, cryptography, combinatorics, and computer science.

## 2.11 Abstract Vectors

Vectors do not have to be columns of numbers.

The following objects may form vector spaces:

| Vector space | Scalars | Example vector |
|---|---|---|
| \(\mathbb{R}^n\) | \(\mathbb{R}\) | Column vectors |
| \(P_n(\mathbb{R})\) | \(\mathbb{R}\) | Polynomials of degree at most \(n\) |
| \(C[a,b]\) | \(\mathbb{R}\) | Continuous functions on \([a,b]\) |
| \(M_{m,n}(\mathbb{R})\) | \(\mathbb{R}\) | \(m \times n\) matrices |
| \(\mathbb{C}^n\) | \(\mathbb{C}\) | Complex coordinate vectors |

For example, the set of polynomials of degree at most \(2\) is a vector space over \(\mathbb{R}\). If

$$
p(x) = 1 + 2x - x^2
$$

and

$$
q(x) = 3 - x + 4x^2,
$$

then

$$
p(x) + q(x) = 4 + x + 3x^2.
$$

If \(c = 2\), then

$$
2p(x) = 2 + 4x - 2x^2.
$$

The vectors here are polynomials, not arrows or columns. The same vector space rules still apply.

## 2.12 Closure

Closure means that an operation stays inside the set.

For vector addition, closure means:

$$
u,v \in V \implies u+v \in V.
$$

For scalar multiplication, closure means:

$$
c \in F,\ v \in V \implies cv \in V.
$$

Closure is essential. If a set fails closure under addition or scalar multiplication, it cannot be a vector space.

For example, the set of vectors in \(\mathbb{R}^2\) whose components are positive is not a vector space. If

$$
v =
\begin{bmatrix}
1 \\
1
\end{bmatrix},
$$

then

$$
-1v =
\begin{bmatrix}
-1 \\
-1
\end{bmatrix},
$$

which has negative components and no longer belongs to the set.

## 2.13 Linear Combinations

A linear combination of vectors \(v_1,\ldots,v_k\) is an expression of the form

$$
c_1v_1 + c_2v_2 + \cdots + c_kv_k,
$$

where \(c_1,\ldots,c_k\) are scalars.

The coefficients come from the field. The result is a vector.

For example, in \(\mathbb{R}^2\),

$$
2
\begin{bmatrix}
1 \\
0
\end{bmatrix}
+
3
\begin{bmatrix}
0 \\
1
\end{bmatrix} =
\begin{bmatrix}
2 \\
3
\end{bmatrix}.
$$

Linear combinations are the basic expressions of linear algebra. Span, independence, basis, dimension, rank, and linear transformations all depend on them.

## 2.14 Why the Field Matters

The field determines which scalar coefficients are allowed.

Consider the equation

$$
2x = 1.
$$

Over \(\mathbb{R}\), the solution is

$$
x = \frac{1}{2}.
$$

Over \(\mathbb{F}_5\), the solution is

$$
x = 3,
$$

because

$$
2 \cdot 3 = 6 \equiv 1 \pmod 5.
$$

The same symbolic equation has different arithmetic in different fields. Linear algebra is therefore always tied to its scalar field.

## 2.15 Summary

Scalars are elements of a field. Vectors are elements of a vector space. A vector space over a field \(F\) is a set of vectors that can be added to each other and multiplied by scalars from \(F\).

The main distinction is this:

| Object | Belongs to | Role |
|---|---|---|
| Scalar | Field \(F\) | Scales vectors |
| Vector | Vector space \(V\) | Object being added and scaled |
| Field | Scalar system | Supplies arithmetic |
| Vector space | Set of vectors | Supplies linear structure |

The phrase “over a field” records which scalars are allowed. Real vector spaces use real scalars. Complex vector spaces use complex scalars. Other fields give other forms of linear algebra.