Kernel of a linear map

A linear map has trivial kernel when its kernel is $\{0\}$. This is equivalent to the map being injective.

For a linear map $T:V\to W$ with finite-dimensional domain,If a basis of $\ker T$ is extended to a basis of $V$, the images of the added basis vectors form a basis of $\mathrm{im}T$. Counting the two parts proves the formula.

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.