The column space of a matrix is the span, or all possible linear combinations, of its columns.

Let’s look at some examples of column spaces and what vectors are in the column space of a matrix. Note that since it is the span of a set of vectors, the column space is itself a vector space.

## Example and discussion

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### Notation

We denote the column space of a matrix as \(\text{Col }A\).

### Example with a 3 x 4 matrix

Let \(A = \left[\begin{array}{cccc} -2 & -1 & 1 & 5\\ 6 & 10 & 0 & -3\\ 7 & 0 & 1 & 0\\ \end{array}\right]\)

The column space of this matrix is:

\(\text{Span}\{\left[\begin{array}{c} -2 \\6 \\ 7\\ \end{array}\right],\left[\begin{array}{c} -1 \\10 \\ 0\\ \end{array}\right],\left[\begin{array}{c} 1 \\0 \\ 1\\ \end{array}\right],\left[\begin{array}{c} 5 \\-3 \\ 0\\ \end{array}\right]\}\)

### What are some vectors in the columns space of A?

Any linear combination of the columns are in the columns space since that is the definition of span from above. This means every column is in the column space. For example, the second column can be written as:

\(\left[\begin{array}{c} -1 \\10 \\ 0\\ \end{array}\right] = (0)\left[\begin{array}{c} -2 \\6 \\ 7\\ \end{array}\right] + (1)\left[\begin{array}{c} -1 \\10 \\ 0\\ \end{array}\right] + (0)\left[\begin{array}{c} 1 \\0 \\ 1\\ \end{array}\right] + (0)\left[\begin{array}{c} 5 \\-3 \\ 0\\ \end{array}\right]\)

The zero vector is in the column space. This can be shown by letting all the weights equal zero. Remember that this must be the case in order for this to be a vector space (well a subspace but we will get that in a minute, anyway any subspace of a vector space is a vector space in its own right.)

\(\left[\begin{array}{c} 0 \\0 \\ 0\\ \end{array}\right] = (0)\left[\begin{array}{c} -2 \\6 \\ 7\\ \end{array}\right] + (0)\left[\begin{array}{c} -1 \\10 \\ 0\\ \end{array}\right] + (0)\left[\begin{array}{c} 1 \\0 \\ 1\\ \end{array}\right] + (0)\left[\begin{array}{c} 5 \\-3 \\ 0\\ \end{array}\right]\)

Finally, to reiterate, any linear combination of these columns are in the column space. So we can use any weights to find such a vector. For example:

\(\left[\begin{array}{c} 1 \\43 \\ 9\\ \end{array}\right] = (1)\left[\begin{array}{c} -2 \\6 \\ 7\\ \end{array}\right] + (4)\left[\begin{array}{c} -1 \\10 \\ 0\\ \end{array}\right] + (2)\left[\begin{array}{c} 1 \\0 \\ 1\\ \end{array}\right] + (1)\left[\begin{array}{c} 5 \\-3 \\ 0\\ \end{array}\right]\)

So:

\(\left[\begin{array}{c} 1 \\43 \\ 9\\ \end{array}\right] \in \text{Col }A\)

### What space is the column space a subspace of?

When you are determining this, count the number of entries in the vectors that make up the columns. Each vector has three entries, so the vectors are in \(\mathbb{R}^3\). That means any linear combination of them is in the same space. Thus the column space is a subspace of \(\mathbb{R}^3\) in this example.

### Other spaces associated with matrices

With a matrix, you can also talk about the row space and the null space.