We got a glimpse of linear transformations when we looked at the
correspondence between elementary row operations and elementary matrices.
We now look at linear transformations more generally.
Let and be vector spaces with scalars coming from the
same field .
A mapping is a linear transformation
if for any two vectors and in and any scalar
, the following are satisfied:
A couple of consequences of this definition are:
for any vectors and
scalars .
where denotes the zero vector in
and denotes the zero vector in .
Examples
Let .
Then is a linear transformation
from to .
To see this, we need to check the two conditions.
Let and .
Then
and .
Let be a mapping from ,
the vector space of polynomials
in with real coefficients having degree at most ,
to given by
.
For example, since
can be written as .
To check that is a linear transformation, take
two vectors and
from and a real number .
Then,
and