Consider the simple example of a matrix:

The matrix has two columns:

The column space, denoted , is the span of these two vectors:

Finding the Column Space

We observe that the two columns and are linearly dependent:

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

Therefore, the column space of is:

which represents all vectors of the form:

In other words, the column space is a line in through the origin in the direction of

Rank of

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