Rank (Linear Algebra)

In linear algebra, the rank of a matrix is the dimension of the vector space spanned by its columns, equivalently the largest number of linearly independent columns the matrix has. This value always equals the dimension of the space spanned by the matrix's rows, a basic equivalence in the subject. Rank measures how non-degenerate a system of linear equations or a linear transformation encoded by the matrix is: a matrix has full rank when its rank equals the smaller of its number of rows or columns, and is called rank deficient otherwise. Only the zero matrix has rank zero, and for a square matrix, having full rank is equivalent to being invertible.