Image and Kernel of a Matrix

Ponder...

For , is the span of the columns and is the set of vectors orthogonal to every row.

Proposition 3.6

The image and kernel of the linear map defined by the matrix are given by

and

Proof

The components are related by . If is the standard basis of , then under ,

so by linearity

Thus , the span of the columns.

For the kernel, the components of the image are . If , then for all , so is the set of vectors orthogonal to all the rows of .

Example 3.7 (Examples of Matrices as Linear Maps)

Zero map. The zero map is defined by taking .
Identity map. The identity map is defined by taking , where is the identity matrix.
Consider the map where . Let be defined by

then, the matrix associated to is

with columns

and rows

Hence, the image and kernel of the linear map are given by

because we have that .

Then, for the kernel, we need

Hence