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From this formula, we obtain
In Section 4.2 we defined the composition of matrix transformations. The following definition extends that concept to
general linear transformations.
DEFINITION (2)
If and are linear transformations, then the composition of with , denoted by
(which is read “ circle ”), is the function defined by the formula
where u is a vector in U.
Remark Observe that this definition requires that the domain of (which is V) contain the range of ; this is essential for
the formula to make sense (Figure 8.1.5). The reader should compare 2 to Formula 18 in Section 4.2.
Figure 8.1.5
The composition of with .
The next result shows that the composition of two linear transformations is itself a linear transformation.
THEOREM 8.1.2
If and are linear transformations, then is also a linear transformation.
Proof If u and v are vectors in U and c is a scalar, then it follows from 2 and the linearity of and that
