So far we have mostly only been considering linear transformations of , even though the definitions are stated for . In this section we will briefly consider linear transformations of , and in particular we would like to understand how to think about them as matrices.
As usual, we will consider the standard basis vectors
Analogous to the 2-dimensional case, we define the scalars for by the equations
and write . So is the matrix associated to the linear transformation .
Conversely, every matrix defines a linear transformation by , via matrix multiplication. In this way, we have a bijective correspondence between linear transformations and matrices, as described in Section 6.2.
Example 6.4.1.
Any translation
is not a linear transformation.
The reflection in any plane is a linear transformation. For instance, the reflection in the -plane, which fixes and sends to is given by the diagonal matrix .