Question: Let A be a $6 \times 3$ matrix. What must $m$ and $n$ be if we define the transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$ by $T(x) = A x$?

1. **State the problem:** We have a matrix $A$ of size $6 \times 3$ and a linear transformation $T$ defined by $T(x) = A x$ where $x \in \mathbf{R}^m$ and $T(x) \in \mathbf{R}^n$. We want to find the values of $m$ and $n$. 2. **Recall the rule for matrix transformations:** If $A$ is an $n \times m$ matrix, then the transformation $T: \mathbf{R}^m \rightarrow \mathbf{R}^n$ defined by $T(x) = A x$ takes vectors from $\mathbf{R}^m$ to $\mathbf{R}^n$. 3. **Apply the rule:** Given $A$ is $6 \times 3$, it has $6$ rows and $3$ columns. 4. **Interpret dimensions:** The number of columns of $A$ must match the dimension of the input vector space $\mathbf{R}^m$, so $m = 3$. 5. **Output dimension:** The number of rows of $A$ determines the dimension of the output space $\mathbf{R}^n$, so $n = 6$. **Final answer:** $$m = 3, \quad n = 6$$