1. Problem: Suppose that $T : \mathbb{R}^5 \to \mathbb{R}^2$ and $T(x) = Ax$ for some matrix $A$ and each $x$ in $\mathbb{R}^5$. How many rows and columns does $A$ have? Step 1: Understand the dimensions involved. - The transformation $T$ maps vectors from $\mathbb{R}^5$ (5-dimensional space) to $\mathbb{R}^2$ (2-dimensional space). Step 2: Recall the rule for matrix dimensions in linear transformations. - If $T(x) = Ax$ where $x \in \mathbb{R}^n$ and $T(x) \in \mathbb{R}^m$, then $A$ is an $m \times n$ matrix. Step 3: Apply the rule. - Here, $n=5$ and $m=2$, so $A$ has $2$ rows and $5$ columns. Final answer: $A$ is a $2 \times 5$ matrix.