Matrix Column Space Calculator

About this calculator

The column space of a matrix is the set of all linear combinations of its columns — geometrically, the subspace that Ax can actually reach as x ranges over all vectors. Its dimension equals the matrix's rank, and a basis for it is given by the original columns of the matrix that correspond to the pivot positions found during Gaussian elimination to row-echelon form (non-pivot columns are redundant, being linear combinations of the pivot ones).

For a 3×3 matrix, this calculator reduces the matrix to echelon form to identify the pivot columns, reports the rank and a basis for the column space, and computes the determinant. The column space determines whether a linear system Ax = b has a solution for a given b (it does exactly when b lies in the column space), which is why this is a core tool in solving systems of equations, checking invertibility, and understanding linear transformations in engineering and computer graphics.