column space

Column space / range

The column space $\text{col}(A)$ of a matrix $A$ is the set of all vectors that can be written as a linear combintation of the columns of $A$.

$$\text{col}(A)=\{y\in {\mathbb{R}}^m\mid y=Ax,x\in {\mathbb{R}}^n\}$$

The column space of a matrix is a subspace of ${\mathbb{R}}^m$.

Example

$$A=\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right],\text{col}(A)=\text{span}\left\{\left[\begin{array}{c}1\\ 2\end{array}\right]\right\}$$

EXAMPLE

The vector $b$ can be represented as a linear combination of columns of $A$:

$$\begin{array}{r}\left[\begin{array}{cc}1& 2\\ 2& 3\\ 3& 4\end{array}\right]\cdot {\left[\begin{array}{c}0\\ 1\end{array}\right]}^T=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right]\end{array}$$

There is no $x$ that can solve the below system of equations, as the column space of $A$ does not contain $b$:

$$\begin{array}{r}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\end{array}\right]\cdot x=\left[\begin{array}{c}1\\ 2\end{array}\right]\end{array}$$