Kernel, Range, Nullity & Rank for this linear transformation

Question

Transformation $T$ from a $2\times 3$ matrix to a $3\times 2$ matrix is defined by $T(A) = A^T$.

My book says that the kernel for such a transformation would only have the $2\times 3$ zero matrix. I've been struggling with this problem. How can I find the the remaining properties for this?

Answer 1

Since every $3\times2$ matrix is the transpose of some $2\times3$ matrix, the range of $T$ is $\mathbb R^{3\times2}$. And $\operatorname{rank}T=\dim\mathbb R^{3\times2}=6$.

Answer 2

using the dimension theorem, $$\forall A \in Mat_{m \times n}:dim\ R(A) = dim \ C(A) = n - Null(A)$$

As $dim R(A) = dim$ row vector space of the matrix $A$, and $dim \ C(A) = dim$ columns vector space $A$.

notice that if $m <n$, then $null(A) > 0$.

for every matrix $A: \ dim(A) = dim(A^T)$