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.
However, this applies to linear transformations as well. 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 since we can relate a matrix with a linear transformation we then obtain:
- dim (im T) + dim (ker T) = dim V
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- rank T + nullity T = dim V.