Now any element in can be expressed as,
Then, the image is the set of all linear combinations,
It is a linear transformation,
But note that,
Correspond to the coloum vectors of the matrix .
Thus, is a space spanned by the linear combinations of the coloum vectors. That means it has a dimension which is called the rank of . Alternatievly, it is the dimension of the set mentioned above, which is the dimension of the image space.