Rank-nullity theorem

T : V \to W

dim V = dim ker T + dim im T .

If a basis of

ker T

is extended to a basis of

V

, the images of the added basis vectors form a basis of

im T

. Counting the two parts proves the formula.

Kernel jump in a parameter-dependent linear map

For a matrix whose entries depend on a parameter, a nonzero maximal minor certifies a fixed matrix rank away from its zeros. At a zero, other minors must still be checked: one vanishing minor does not itself imply a rank drop. Once the matrix rank is determined, the rank-nullity theorem gives the dimension of the kernel of a linear map. Exceptional parameters can then be solved separately to exhibit the additional kernel of a linear map directions.

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The Rank-Nullity Theorem is a fundamental result in linear algebra that relates the dimensions of different subspaces associated with a linear transformation. Specifically, it applies to linear transformations between finite-dimensional vector spaces.