A matrix's rank counts its truly independent directions
Line up the columns of a matrix and ask how many genuinely point in new directions rather than being combinations of the others. That count is the rank. Surprisingly, asking the same question of the rows always gives exactly the same number, one of the foundational facts of linear algebra.
Formally, the rank is the dimension of the space spanned by a matrix's columns, equivalently the largest number of linearly independent columns. It measures how nondegenerate the system of equations or transformation encoded by the matrix is. Written rank(A) or rk(A), it sometimes appears as rg(A), from the German Rang. For a general linear map between vector spaces, the rank is the dimension of its image.
Consider a three-by-three matrix whose third column equals the first plus the second. The first two are independent, so the rank is at least 2, but the dependent third column keeps it below 3. A matrix reaches full rank when its rank equals the smaller of its row and column counts; anything less is rank-deficient, and the shortfall is the rank deficiency. Because a matrix's columns are its transpose's rows, row rank equaling column rank also means a matrix and its transpose share the same rank.
By hand, the usual method is Gaussian elimination into row echelon form, since row operations change neither rank. The rank then equals the number of pivots, or the number of rows that are not all zeros. Computers, however, struggle: in floating-point arithmetic basic elimination can mislead, so rank-revealing methods are preferred, such as singular value decomposition or the cheaper QR factorisation with pivoting. Even then someone must decide when a tiny value should count as zero, a judgement depending on the matrix and its use.
Many proofs exist for the equality of row and column rank. One, based on Wardlaw's 2005 work, relies only on linear combinations and holds over any field. Another, following Mackiw's 1995 paper, uses orthogonality and works for real matrices. The rank–nullity theorem supplies yet another definition, via the dimension of the kernel.
Source: Rank (linear algebra)
