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


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

Explore with Wolfram|Alpha

Cite this as:

Jackson, Rahmi. "Rank-Nullity Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications