Column Space

Definition

The column space of a matrix, $A$, often denoted by $C(A)$, is a vector space spanned by the column vectors of $A$.

The column-space of an $m \times n$ matrix $A$ is a subspace of $\R^m$. The dimensions of the column space are the number of linearly independent columns, that is, the rank of the matrix $A$.

Examples

1. The column space of $\begin{bmatrix}1 &0\\0&1\end{bmatrix}$ is $\R^2$.

2. The column space of $\begin{bmatrix}1 &2\\2&4\end{bmatrix}$ is a one-dimensional subspace of $\R^2$. This is because there’s a single linearly independent vector as $2\bold{c_1} =\bold{c_2}$. Any of these columns can be a basis for the subspace.

3. The column space of $\begin{bmatrix}0 &0\\0&0\end{bmatrix}$ is a zero-dimensional subspace of $\R^2$.

4. The column space of an $n \times n$ invertible matrix $A$ is $\R^n$ ...