Students will learn that the column space of a matrix, is the span of all its vectors. For example, if \(A\) is an \(m \times n\) matrix, where

$$A = [\vec{v_1}, \vec{v_1}, \ldots \vec{v_n}]$$Then it's column space is

$$C(A) = \text{span}(\vec{v_1}, \vec{v_1}, \ldots \vec{v_n})$$Next, students will learn why the column space of a matrix is closed under scalar multiplication and vector addition. That is, if \(\vec{a}, \vec{b} \in C(A),\) then \(s\vec{a} \in C(A),\) and \(\vec{a} + \vec{b} \in C(A).\) Finally, students will learn that saying there is no solution to the system \(A\vec{x} = \vec{b},\) is equivalent to saying \(\vec{b} \not\in C(A).\) Here's a lesson on all that.