Definition of vector

Definition: A vector is an ordered list of (n) scalars from a field (\mathbb{F}).

A vector in (\mathbb{F}^n) can be written as:

[\mathbf{v} = (v_1, v_2, \ldots, v_n)]

where each (v_j \in \mathbb{F}) is called a component (or entry) of the vector.

Notation: Vectors are typically denoted by boldface lowercase letters: (\mathbf{u}), (\mathbf{v}), (\mathbf{w}), etc.

Two common representations:

• Row vector: (\mathbf{v} = (v_1, v_2, \ldots, v_n)) — written horizontally
• Column vector: (\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}) — written vertically

Examples:

• (\mathbf{v} = (3, -1, 0) \in \mathbb{R}^3)
• (\mathbf{w} = (1+i, 2-i) \in \mathbb{C}^2)