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

• Vector addition: $V\times V\to V$
• Scalar multiplication: $F\times V\to V$

satisfying the following axioms:

1. $(V,+)$ is an abelian group

1. Scalar multiplication is associative: $a(bv)=(ab)v$

1. Distributive laws hold

1. Identity: $1v=v$ for all $v\in V$

A scalar is an element of a field used to define a vector space.

A vector is an element in a vector space.

A vector whose endpoints are not fixed at particular points is called a free vector.

A vector with a fixed endpoint is called a bound vector.

Let $\mathbf{x}=(x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^n$. The magnitude, or length, $\mathbf{x}$ is denoted as $||\mathbf{x}||$ and is defined as:

$$||\mathbf{x}||=\sqrt{{x_1}^2+{x_2}^2+\cdots +{x_n}^2}=\sqrt{\mathbf{x}\cdot \mathbf{x}}=\sqrt{\sum _{i=1}^n{x}_i^2}$$

The direction of a vector can be specified by the angle between it and some fixed reference, such as the $x$-axis.

The zero vector $(0,0,\cdots ,0)$ is denoted as $\mathbf{0}$ and has no direction.

Two vectors are equal if they have the same coordinates (or equivalently, the same length and direction).

If a vector has a length of 1, i.e. if $||x||=1$, we say the vector is a unit vector.

We perform vector addition by adding two vectors $\mathbf{x}=(x_1,x_2,\cdots ,x_n)$ and $\mathbf{y}=(y_1,y_2,\cdots ,y_n)$ according to the following rule:

$$\mathbf{x}+\mathbf{y}=(x_1+y_1,x_2+y_2,\cdots ,x_n+y_n)$$

that is, by making a new vector where the coordinates are the sums of the respective coordinates in the vectors being summed.

Similarly, vector subtraction can be performed as:

$$\mathbf{x}-\mathbf{y}=(x_1-y_1,x_2-y_2,\cdots ,x_n-y_n)$$

We can multiply vectors by a scalar. Given the scalar $c$:

$$c\mathbf{x}=(c{x}_1,c{x}_2,\cdots ,c{x}_n)$$

If $\mathbf{x}$ and $\mathbf{y}$ are vectors in ${\mathbb{R}}^n,$ then their inner product is defined as

$$\mathbf{X}\cdot \mathbf{y}=\sum _{i=1}^n{x}_i{y}_i.$$

Two vectors $\mathbf{u}$ and $\mathbf{v}$ are said to be perpendicular or orthogonal if the angle between them is $\pi /2$ radians, or, equivalently, if the inner product $\mathbf{u}\cdot \mathbf{v}=0.$

Two vectors $\mathbf{x}$ and $\mathbf{y}$ are said to be parallel if $\mathbf{x}$ is a scalar multiple of $\mathbf{y}$, i.e., if there exists some scalar $c$ where $\mathbf{x}=c\mathbf{y}.$

The cross product $\mathbf{a}\times \mathbf{b}$ (read "a cross b") of two vectors $\mathbf{a}$ and $\mathbf{b}$ is the vector $\mathbf{v}$ denoted by

$$\mathbf{v}=\mathbf{a}\times \mathbf{b}.$$

I. If $\mathbf{a}=\mathbf{0}$ or $\mathbf{b}=\mathbf{0},$ then we define $\mathbf{v}=\mathbf{a}\times \mathbf{b}=\mathbf{0}.$

II. If both vectors are nonzero vectors, then vector $\mathbf{v}$ has the length

$$\begin{array}{cc}|\mathbf{v}|=|\mathbf{a}\times \mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin \theta ,& \text{(1)}\end{array}$$

where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}.$ Furthermore, $\mathbf{a}$ and $\mathbf{b}$ form the sides of a parallelogram on a plane in space. The area of this parallelogram is precisely given by (1), such that the length $|\mathbf{v}|$ of the vector $\mathbf{v}$ is equal to the area of the parallelogram.

III. If $\mathbf{a}$ and $\mathbf{b}$ lie in the same straight line, i.e. $\mathbf{a}$ and $\mathbf{b}$ have the same or opposite directions, then $\theta$ is $0^{\circ }$ or ${180}^{\circ }$ so that $\sin \theta =0.$ In that case $|\mathbf{v}|=0,$ so that $\mathbf{v}=\mathbf{a}\times \mathbf{b}=\mathbf{0}.$

IV. If cases I and III do not occur, then $\mathbf{v}$ is a nonzero vector. The direction of $\mathbf{v}=\mathbf{a}\times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ such that $\mathbf{a},\mathbf{b},\mathbf{v},$ precisely in this order, form a right-handed triple

Let ${\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},\cdots ,{\mathbf{v}}_{\mathbf{n}}\in {\mathbb{R}}^n$ and $c_1,c_2,\cdots ,c_3\in \mathbb{R}.$ Then, the vector

$$\mathbf{v}=c_1{\mathbf{v}}_{\mathbf{1}}+c_2{\mathbf{v}}_{\mathbf{2}}+\cdots +c_n{\mathbf{v}}_{\mathbf{n}}$$

is called a linear combination of ${\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},\cdots ,{\mathbf{v}}_{\mathbf{n}}.$

Let ${\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},\cdots ,{\mathbf{v}}_{\mathbf{n}}\in {\mathbb{R}}^n.$ The set of all linear combinations of ${\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},\cdots ,{\mathbf{v}}_{\mathbf{n}}$ is called their span, denoted $\mathrm{Span}({\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},\cdots ,{\mathbf{v}}_{\mathbf{n}}).$

That is:

$$\mathrm{Span}({\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},\cdots ,{\mathbf{v}}_{\mathbf{n}})=\{\mathbf{v}\in {\mathbb{R}}^n:\mathbf{v}=c_1{\mathbf{v}}_{\mathbf{1}}+c_2{\mathbf{v}}_{\mathbf{2}}+\cdots +c_n{\mathbf{v}}_{\mathbf{n}}\text{for some scalars}{c}_1,c_2,\cdots ,c_n\}$$