Rankkritein

Rankkritein is a term used in Finnish mathematical literature to refer to a criterion or condition by which the rank of a matrix or linear transformation is determined. In linear algebra, the rank of a matrix is the dimension of its row space (equivalently, its column space) and equals the maximum number of linearly independent rows or columns. Rankkritein thus provides concrete checks or methods to establish this quantity.

Common ways to determine rank include row reducing a matrix to row echelon form or reduced row echelon form and counting the pivots, or examining the number of nonzero singular values through a singular value decomposition. For a square matrix, a full rank condition is equivalent to having a nonzero determinant. In general, the rank can also be characterized by the existence of invertible submatrices of a given size, and by the dimension formula for row and column spaces.

Rank criteria are central to solving linear systems. The system A x = b is solvable if and only if the augmented matrix [A|b] has the same rank as A. The solution is unique when the rank of A equals the number of unknowns, otherwise there are infinitely many solutions if the rank is less than the number of unknowns, or no solution if the ranks differ between A and [A|b].

Properties related to rank include the inequality rank(AB) ≤ min(rank(A), rank(B)) and the fact that rank(A^T) = rank(A). Computationally, numerical rank introduces a tolerance to determine which singular values are treated as effectively zero.

In summary, rankkritein encompasses the set of criteria used to determine or reason about the rank of matrices and linear maps, with wide applications in solving systems, analyzing invertibility, and understanding linear dependence.