For a linear map $T:V\to W$ with finite-dimensional domain, rank is $\dim\operatorname{range}T$ and nullity is $\dim\ker T$. The rank–nullity theorem gives $\operatorname{rank}T+\operatorname{nullity}T=\dim V$. For an $m\times n$ matrix of rank $r$, right nullity is $n-r$, while left nullity is $m-r$.
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- Definition of rank of a linear transformationThe rank of $T:V\to W$ is the dimension of its image, $\operatorname{rank}T=\dim T(V)$. For a matrix representation...
- Definition of nullity of a linear transformationThe nullity of $T:V\to W$ is $\dim\ker T$, the number of independent directions mapped to zero. For a...
- A linear transformation is surjective if and only if the columns of its matrix span the codomain.Let $A=[T] {C\leftarrow B}$. The map $T:V\to W$ is surjective iff its matrix columns span the entire coordinate space...
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