The column space of a matrix is a vector space

**Theorem:** The *column space* of a matrix \(A\) (the span of its column vectors) is a vector space --- specifically, a subspace of \(\mathbb{F}^m\) where \(m\) is the number of rows.

Notation: \(\text{col}(A)\), \(\text{colspace}(A)\), or \(\text{im}(A)\).

The column space equals the range of the linear transformation \(T(\mathbf{x}) = A\mathbf{x}\).

**Key fact:** The dimension of the column space is the *rank* of \(A\). Row rank equals column rank.

**Example:** For \(A = \begin{pmatrix} 1 & 2 \ 3 & 6 \ 0 & 0 \end{pmatrix}\), the column space is spanned by \(\begin{pmatrix} 1 \ 3 \ 0 \end{pmatrix}\) (since column 2 = 2 times column 1), a 1-dimensional subspace of \(\mathbb{R}^3\).Copy