Overview
The column space of a matrix {% A %} is the vector space spanned by the columns of the matrix.
More formally, the column space of a {% m \times n %} matrix such that
{% \mathcal{C}(A) = \{ \vec{y} \in \mathbb{R}^n | \exists \vec{x} \in \mathbb{R}^m such \; that \; \vec{y} = A \vec{x} \} %}
Properties
- {% \mathcal{C}(A) = \mathcal{C}(AA^T) %}
- {% \mathcal{C}(A) ^\perp = \mathcal{C}(A^\perp) = Kernel(A^T) %}
- {% \mathcal{C}(A) \subset \mathcal{C}(B) \longleftrightarrow \mathcal{C}(B)^\perp \subset \mathcal{C}(A)^\perp %}