Column Space - Yousef's Notes

The column space of a matrix $A$ is the set of all linear combinations of the columns of $A$. Given a matrix $A$ with columns $\mathbf{a_1}, \mathbf{a_2}, \ldots, \mathbf{a_n}$, the column space of $A$ is denoted as $\col(A)$ or $C(A)$. The column space can be thought of as the Span of the columns of $A$, and it represents the set of all possible linear combinations of the columns. For example, consider a $2 \times 2$ matrix: $$ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$ The column space of $A$ is the set of all linear combinations of the columns: $$ \col(A) = \{c_1\begin{pmatrix} 1 \\ 3 \end{pmatrix} + c_2\begin{pmatrix} 2 \\ 4 \end{pmatrix} : c_1, c_2 \in \mathbb{R}\} $$

The column space has the following properties:

Subspace: The column space is a subspace of the ambient space (e.g., $\mathbb{R}^2$ or $\mathbb{R}^3$).
Span: The column space is the span of the columns of $A$.
Basis: The columns of $A$ form a basis for the column space if and only if they are linearly independent. The column space is important in many applications, including:
Linear Transformation: The column space represents the range of the linear transformation represented by $A$.
Solving systems of linear equations: The column space is used to determine the consistency of a system of linear equations.
Least squares: The column space is used in least squares regression to find the best-fitting line or hyperplane.