## FANDOM

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The rank theorem is a theorem in linear algebra that states that the rank of a matrix $A$ plus the dimension of the null space of $A$ will be equal to the number of columns of $A$ .

$n=\text{rank}(A)+\dim\bigl(\text{null}(A)\bigr)$

Since the rank is equal to the dimension of the image space (or column space, since they are identical, and the row space since the dimension of the row space and column space are equal) this can be expanded to

\begin{align}n&=\text{rank}(A)+\dim\bigl(\text{null}(A)\bigr)\\ &=\dim\bigl(\text{im}(A)\bigr)+\dim\bigl(\text{null}(A)\bigr)\\ &=\dim\bigl(\text{col}(A)\bigr)+\dim\bigl(\text{null}(A)\bigr)\\ &=\dim\bigl(\text{row}(A)\bigr)+\dim\bigl(\text{null}(A)\bigr)\end{align}