Column Space

It is the set of vectors the are outputs of $A\vec{x}$.

$$x_1{\vec{a}}_1+x_2{\vec{a}}_2+...$$

This is literally a linear combination of the the columns of $A$. Where $x_n$ are the coefficients.

• So the columns space is literally the span of the columns of $A$.
• By definition, columns space is a subset of $R^m$.

Theorem 28.4

• $\text{Col}(A)$ is a subspace of $R^m$. $m$ not $n$.
• $\text{Col}(A)$ is the set of linear combinations of the columns of the matrix.
• Each column of $A$ is $m$ entries tall cuz $A\in M_{m\times n}(\mathbb{R})$ ($A$ has $m$ rows).