Vector

linear-algebra

Definition

Vector

Mathematical (Element of a Vector Space) vector is an element $v$ of a vector space $V$ over a field $F$ — an object on which the space's operations of addition and scalar multiplication act, and which carries no further structure beyond what $V$ prescribes.

A

Physics (Magnitude and Direction) vector is a geometric quantity endowed with magnitude and direction, representable as an arrow in space. Two vectors are equal when they have the same magnitude and direction, regardless of where their tails sit, so the arrow can be translated freely.

A

Computer Science (List of Components) vector is a finite ordered list of scalars from a field $F$, written as a column or row:

A

$$\mathbf{v}=\underset{\text{Column Vector}}{\underbrace{\left[\begin{array}{c}{\mathbf{v}}_1\\ {\mathbf{v}}_2\\ ⋮\\ {\mathbf{v}}_n\end{array}\right]}}=\underset{\text{Row Vector}}{\underbrace{\left[\begin{array}{cccc}{\mathbf{v}}_1& {\mathbf{v}}_2& \dots & {\mathbf{v}}_n\end{array}\right]}},$$

where each ${\mathbf{v}}_i\in F$ is a component. The two layouts are interchangeable representations of the same data.