Rank-nullity theorem
In linear algebra, the rank-nullity theorem, in its simplest form, relates the rank and the nullity of a matrix together with the number of columns of the matrix. Specifically, if A is an m-by-n matrix over the field F, then
- rank A + nullity A = n.
However, this applies to linear transformations as well. Let V and W be vector spaces over the field F and let T : V → W be a linear transformation. Then the rank of T is the dimension of the image of T, the nullity the dimension of the kernel of T, and we have
- dim (im T) + dim (ker T) = dim V
thus, equivalently,
- rank T + nullity T = dim V.
This is in fact more general than the matrix statement above, because here V and W may even be infinite-dimensional.
To prove the theorem, one starts with a basis of the kernel of T, and extends it to a basis of all of V. It is then not too difficult to show that T applied to the "new" basis vectors yields a basis of the image of T.
In more modern language, the theorem can also be phrased as follows: if
- 0 → U → V → R → 0
is a short exact sequence of vector spaces, then
- dim(U) + dim(R) = dim(V)
Here R plays the role of im T and U is ker T.
