Rank-nullity theorem
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
thus, equivalently,
- rank T + nullity T = dim V.
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List of linear algebra topics | List of mathematical topics (P-R)
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