Fundamental Linear Transformation Subspaces

Let be a linear transformation. The kernel of is:

(see set builder notation)

The kernel is also the null space, and in fact is sometimes called the null space of , denoted by .

Find a basis for , where is the linear transformation defined by:

We have:

(see standard matrix)

Carrying to RREF:

To find a basis for the kernel (or null space), we want to set , and find which vectors have free variables:

Which gives:

so the basis for the kernel is

Let be a linear transformation. The range of is:

The range is the same in concept to the column space.

Recall that the column space was the set of all possible outputs of . In this case, the range is the set of all possible outputs of . Also recall that a linear operator is the same as the matrix-vector product. That means these are the same thing.

Another way to think of range is the range of a function, which is defined in the same way. The range of is .

Find a basis for as defined in the previous example:

The basis for the column space is the elements of the matrix that have a leading 1. Think about why this is the case.