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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