Rank, Inner Product and Nonsingularity
It is shown that the column rank of a matrix equals its row rank. Basic properties of rank of a product and sum are proved. Inner product is defined and Gram–Schmidt process is described. Orthogonal projection of a vector and orthogonal complement of a subspace are introduced. the rank plus nullity theorem is proved. Various criteria for nonsingularity are given. The Frobenius inequality for rank is proved.