Definition of column vector

Definition: A column vector is a vector written as a vertical array of (n) entries:

[\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{pmatrix}]

where each (v_j \in \mathbb{F}).

Notation: Column vectors are often denoted using parentheses or square brackets:

[\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}]

Why column vectors? Column vectors are the standard representation in linear algebra because:

• They align naturally with matrix-vector multiplication (A\mathbf{x})
• They correspond to elements of (\mathbb{F}^n) as column spaces

Example:

[\mathbf{v} = \begin{pmatrix} 3 \\ -1 \\ 0 \end{pmatrix} \in \mathbb{R}^3]

The transpose of a column vector is a row vector: (\mathbf{v}^T = (3, -1, 0)).