Rank (linear algebra)

The rank of a matrix is the dimension of its row space, equivalently its column space. It measures how many independent directions the matrix represents.

Gaussian elimination: A sequence of elementary row operations that transforms a matrix into echelon form. Its pivots identify the independent rows that determine a matrix’s rank.
Linear independence: A set of vectors is linearly independent when no nontrivial combination of them equals zero. Rank counts the independent rows or columns rather than their total number.
Linear system: A collection of linear equations sharing the same variables. The coefficient matrix’s rank helps determine whether the system has solutions and whether they are unique.
Nullity: The dimension of the null space of a linear map or matrix. Rank measures the image dimension; nullity measures the dimension of inputs mapped to zero.
Row echelon form: A matrix form in which nonzero rows precede zero rows and each leading entry lies to the right of the one above. The number of nonzero rows in this form equals the original matrix’s rank.
Vector space: A set with vector addition and scalar multiplication satisfying the vector-space axioms. Row and column spaces are subspaces whose dimensions define rank.
Invertible matrix: A square matrix that has a two-sided multiplicative inverse. A square matrix is invertible exactly when its rank equals its size.
Determinant: A scalar associated with a square matrix that encodes volume scaling and invertibility. A nonzero determinant detects full rank only for square matrices.
Pivot (linear algebra): A leading nonzero entry in a row-echelon matrix, marking a basic variable or independent column. Each pivot contributes one to the rank.
Basis (linear algebra): A linearly independent set of vectors that spans a vector space. A basis of either matrix space has as many elements as the matrix’s rank.

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