- Let $A$ be an $m \times n$ matrix. Suppose that the matrix equation $AX = Y$ is consistent for any $Y$ that is an element of $R^m$ . (a) What is the range of $T_A$? Justify your answer. (b) Under the assumptions above, is it possible that m > n? Justify your answer.
For part (a), I said the range of $T_A$ is $R^m$ because the codomain $Y$ are all possible solutions to the linear transformation $T_A$ with vectors $X$, then $R^m$ contains both consistent and nonconsistent solutions to all linear transformations.
For part (b), I wrote $m > n$ is possible, but I'm having a hard time explaining so. It is fairly obvious because the matrix $A$ is the transformation $T: R^n \rightarrow R^m$. Using the formula $\mathrm{rank}(A) + \mathrm{null}(A) = n$, then $m \geq n$ . Is that a sufficient answer? Any help is welcome.