Rank nullity

• How to Find the Dimension of the Kernel of a Square Matrix
  For an n × n matrix, the kernel has dimension n minus the matrix rank. Row-reduction gives the rank and therefore the nullity.
• How to Prove a Homogeneous System Has Infinitely Many Solutions
  For a homogeneous system with n unknowns and m equations over an infinite field, rank-nullity guarantees a nonzero solution and infinitely many scalar multiples.