Linear Transformation Products
Why
We can consider function composition for linear maps?
Definition
Given vector spaces $U, V, W$ over the same
field $\F $ and linear maps $T \in
\mathcal{L} (U, V)$ and $S \in \mathcal{L} (V,
W)$, the product of $S$
and $T$ is the linear map $R \in
\mathcal{L} (U, W)$ defined by
\[
R(u) = S(T(u)) \quad \text{for all } u \in U
\]
This definition only makes sense if $T$ maps into the domain of $S$. We often say that the maps are conforming in this case.
Notation
Often the product is denoted $ST$ (instead of $S \circ T$).
Algebraic properties
Suppose $T_1, T_2, T_3$ are three linear maps
so that conforming for $T_1T_2T_3$.
Then
\[ (T_1T_2)T_3 = T_1(T_2T_3) \]
Not commutative
Image of zero
Suppose $T$ is a linear map from $V$ to $W$.
Then $T(0) = 0$.