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Rank-Nullity Theorem


Let V and W be vector spaces over a field F, and let T:V->W be a linear transformation. Assuming the dimension of V is finite, then

 dim(V)=dim(Ker(T))+dim(Im(T)),

where dim(V) is the dimension of V, Ker is the kernel, and Im is the image.

Note that dim(Ker(T)) is called the nullity of T and dim(Im(T)) is called the rank of T.


See also

Kernel, Null Space, Nullity, Rank

This entry contributed by Rahmi Jackson

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Cite this as:

Jackson, Rahmi. "Rank-Nullity Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Rank-NullityTheorem.html

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