The statement is incorrect. Let's break it down:

Given:

• Matrix $A$ is $m \times n$,
• $T: \mathbb{R}^n \to \mathbb{R}^m$, defined by $T(x) = A x$.

The condition "for every $b \in \mathbb{R}^m$ in the range of $T$, there is at most one $x \in \mathbb{R}^n$ such that $A x = b$" is saying that the transformation is injective (one-to-one), meaning there are no two distinct $x$'s that map to the same $b$.

Does injectivity imply surjectivity (onto)?

• Injectivity means that each $b$ in the range of $T$ has at most one corresponding $x$.
• **Surject