Let's analyze each statement given the transformations $S : \mathbb{R}^n \rightarrow \mathbb{R}^m$ induced by the matrix $A$ and $T : \mathbb{R}^k \rightarrow \mathbb{R}^n$ induced by the matrix $B$.

1. Statement 1: $A$ is $n \times m$, $B$ is $k \times n$, and $(S \circ T)(x) = B A x$.

   • If $S \circ T$ is the composition of $S$ and $T$, then $(S \circ T)(x) = S(T(x))$. Since $S$ is represented by matrix $A$ and $T$ by matrix $B$, the composition $S(T(x))$ should correspond to applying $T$ first (via matrix $B$) and then applying $S$ (via matrix $A$).
   • For the dimensions to match, $A$ should be $m \times n$, not $n \times m$, because $S$ maps $\mathbb{R}^n \rightarrow \mathbb{R}^m$. Similarly, $B$ should be $n \times k$, not $k \times n$, because $T$ maps $\mathbb{R}^k \rightarrow \mathbb{R}^n$.
   • Therefore, this statement is false.

2. Statement 2: $(S \circ T)(x) = A(Bx)$.

   • This is equivalent to saying that the composition $S(T(x))$ can be represented by applying $T$ first (resulting in $Bx$) and then applying $S$ (resulting in $A(Bx)$).
   • This is a correct representation of the composition $S(T(x))$, so this statement is true.

3. Statement 3: $(S \circ T)$ is determined by applying $S$ first, followed by $T$.

   • This statement is incorrect because, in the composition $S \circ T$, the function $T$ is applied first, followed by $S$. Thus, this statement is false.

4. Statement 4: $(S \circ T)(x) = S(T(x))$.

   • This is the correct definition of the composition $S \circ T$, which states that we apply $T$ to $x$ first and then apply $S$ to the result. Thus, this statement is true.

Summary of True/False Statements:

• Statement 1: False
• Statement 2: True
• Statement 3: False
• Statement 4: True

Would you like a more detailed explanation of any of these steps?

Here are five related questions to explore further:

1. What are the requirements for two matrices to be multiplied in terms of their dimensions?
2. How do compositions of linear transformations correspond to matrix multiplication?
3. Why is the order of matrix multiplication important in compositions of linear transformations?
4. What are some common mistakes to avoid when interpreting compositions of transformations?
5. How can we represent more complex compositions of transformations in matrix form?

Tip: Always verify the dimensions of matrices when performing matrix multiplications or compositions of transformations to ensure compatibility.