rank-nullity theorem clarification

Question

The rank-nullity theorem says that if linear transformation $T: V\rightarrow W$ then $$ \\ \dim(Im(T))+\dim(Ker(T))=\dim(V)$$ and equivalently, $$\\ rank(T) + nullity(T) = \dim(V) $$ when seeing $T$ as a $m \times n $ matrix.

This makes perfect sense to me whenever $V$ and $W$ are vector spaces of column vectors. However, in terms of vector space of matrices, I got confused easily.

The question is: does rank of $T$ always equal to the dimension of its image? Also how about the relationship between nullity and $\dim (ker(T))$?

Say if $V$ is the vector space of all $2 \times 3$ matrices, and $W$ is the vector space of all $4 \times 1$ matrices, and if T is onto, then what would T look like? (i.e. what are m and n?)

Following the first expression of RNT, it suggests that T should have a corresponding range of 4, and the kernel should have a dimension of 2, but does it mean that $rank(T) = 4$ and $nullity(T) = 2$? If yes, then shouldn't $T$ have definitely 2 columns since $V$ is $2 \times 3$?

It is really hard to figure out what you are asking. If the first two statements are equivalent, then you have the answer to your first two questions. — copper.hat, Oct 16 '14 at 18:14
@copper.hat sorry I think I made a mistake there. rank(T) + nullity(T) = dim(V) but not n... so then T should be a $4 \times 2$ matrix right? — MatheMagic, Oct 16 '14 at 18:37
$T$ is a map from $\mathbb{R}^{\dim V} \to \mathbb{R}^{\dim W}$. If $V$ is the space of $2 \times 3$ matrices, then $\dim V = 6$. — copper.hat, Oct 16 '14 at 19:32
@copper.hat and that is simply because $ \mathbb{R}^6$ and $\mathbb{R}^4$ is isomorphic to V and W respectively? — MatheMagic, Oct 16 '14 at 19:44
Well, it is because any basis has 6 elements. One basis is $e_i e_j^T$, $i=1,2$, $j=1,2,3$, with the $e_k$ the standard bases in $\mathbb{R}^2$, $\mathbb{R}^3$ resp. — copper.hat, Oct 16 '14 at 20:51
Possible duplicate of Does the rank-nullity theorem hold for infinite dimensional $V$? — Gaurang Tandon, Apr 18 '19 at 11:23

rank-nullity theorem clarification

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