If S and T are transformattion mappings, what is [ST]?

Question

S and T are transformation mappings, what does [ST] and [TS] mean?

Does it mean transform via S and then apply T to the result and vice versa?

Answer 1

Juxtaposition of linear transformations means composition. In details, $ST$ means $S \circ T$ and $TS$ means $T \circ S$ (whenever they are defined, surely). This notation is used thinking of the following: $$T: V_1 \to V_2 \qquad S:V_2 \to V_3 \qquad S\circ T: V_1 \to V_3$$ Let $B_1, B_2, B_3$ be basis for the respective spaces (in finite dimension). Then: $$[S \circ T]_{B_1, B_3} = [S]_{B_2, B_3}[T]_{B_1,B_2}$$ At the bottom of our hearts, we know the distinction, so, writing, we oftem "confuse" the matrix with the transformation, so $S \circ T$ becomes $ST$. Actually, the identity above is the motivation for the definition of matrix multiplication.