Determinants, Rank

Rank

If the rows of a matrix are vectors in a vector space, the span of a matrix is the subspace spanned by its rows, and the rank of a matrix is the dimension of that subspace. The matrix need not be square. If it is an m×n matrix, the rank cannot exceed n; the vectors live in n-dimensional space after all. Nor can the rank exceed m, the number of vectors in the spanning set. The rank is equal to m iff the vectors form a linearly independent set. Let's see how gaussian elimination can help calculate the rank.

Swapping two rows doesn't change the span of the matrix, and neither does row subtraction. If v₂ is replaced with v₂-k×v₁, we can still span the original v₂, simply by adding k×v₁ back again. The span of the matrix is unchanged by gaussian elimination.

Suppose gaussian elimination leaves us with a dependent set of vectors. Thus x₁×r₁ + x₂×r₂ + … x_m×r_m = 0, where x_i is the i^th coefficient and r_i is the i^th nonzero row, or vector. The first nonzero entry in the first row has all zeros below it, hence x₁ has to be 0. That row does not participate. Find the first nonzero entry in the next row. It has all zeros below it, hence x₂ = 0. This continues all the way to x_m, hence the rows are linearly independent. They form a basis for the subspace spanned by the matrix, and the number of rows gives the rank. If the rank is n, the matrix is nonsingular, and spans the entire space.