Column Space

66. Column Space

The columns space (or range) of matrix $𝐴$ is span of its columns vectors

If the matrix $𝐴$ has columns ${\vec{𝑎}}_{1}, {\vec{𝑎}}_{2}, \dots, {\vec{𝑎}}_{𝑛}$ , then the column space of $𝐴$ is defined as:

Col (𝐴) = {\vec{𝑦} \in ℝ^{𝑚} | \vec{𝑦} = 𝐴 \vec{𝑥} for some \vec{𝑥} \in ℝ^{𝑛}}

or equivalently,

Col (𝐴) = span ({{\vec{𝑎}}_{1}, {\vec{𝑎}}_{2}, \dots, {\vec{𝑎}}_{𝑛}})

Example

Consider the simple example of a $2 \times 2$ matrix:

𝐴 = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}]

The matrix has two columns:

{\vec{𝑎}}_{1} = [\begin{matrix} 1 \\ 3 \end{matrix}] and {\vec{𝑎}}_{2} = [\begin{matrix} 2 \\ 6 \end{matrix}]

The column space, denoted $Col (𝐴)$ , is the span of these two vectors:

Col (𝐴) = span ({[\begin{matrix} 1 \\ 3 \end{matrix}], [\begin{matrix} 2 \\ 6 \end{matrix}]})

Finding the Column Space

We observe that the two columns ${\vec{𝑎}}_{1}$ and ${\vec{𝑎}}_{2}$ are linearly dependent:

{\vec{𝑎}}_{2} = 𝑘 {\vec{𝑎}}_{1}

This means that ${\vec{𝑎}}_{2}$ is a scalar multiple of ${\vec{𝑎}}_{1}$ , the the two columns are linearly dependent. As a result, the column space is spanned by just one vector, ${\vec{𝑎}}_{1}$ , because any linear combination of ${\vec{𝑎}}_{1}$ and ${\vec{𝑎}}_{2}$ can be reduced to a multiple of ${\vec{𝑎}}_{1}$ .

Therefore, the column space of $𝐴$ is:

Col (𝐴) = span ({[\begin{matrix} 1 \\ 3 \end{matrix}]})

which represents all vectors of the form:

𝑐 [\begin{matrix} 1 \\ 3 \end{matrix}] = [\begin{matrix} 𝑐 \\ 3 𝑐 \end{matrix}] for any scalar 𝑐

In other words, the column space is a line in $ℝ^{2}$ through the origin in the direction of

Rank of $𝐴$

The rank of $𝐴$ , which is the dimension of its column space, is 1 because there is only one linearly independent column

This means the column space is the span of the columns of $𝐴$ , or all vectors that can be formed by taking linear combinations of the columns of $𝐴$ .