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: \(\mathbf{v}^T = (3, -1, 0)\).
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