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Rank (linear algebra) In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4]
The column space of an m × n matrix with components from is a linear subspace of the m -space . The dimension of the column space is called the rank of the matrix and is at most min (m, n). [1] A definition for matrices over a ring is also possible. The row space is defined similarly.
Matrix (mathematics) An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of ...
As used in linear algebra, an augmented matrix is used to represent the coefficients and the solution vector of each equation set. For the set of equations the coefficients and constant terms give the matrices and hence give the augmented matrix. Note that the rank of the coefficient matrix, which is 3, equals the rank of the augmented matrix ...
Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero.
Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and. the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
Rank factorization. In mathematics, given a field , nonnegative integers , and a matrix , a rank decomposition or rank factorization of A is a factorization of A of the form A = CF, where and , where is the rank of .
Matrix equivalence is an equivalence relation on the space of rectangular matrices. For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions. The matrices can be transformed into one another by a combination of elementary row and column operations. Two matrices are equivalent if and ...