Null Space Calculator

The null space (or kernel) of an m × n matrix A consists of all column vectors x in Rⁿ such that Ax = 0. To find this set, the calculator applies Gaussian elimination to reduce matrix A into its Row Echelon Form (REF) or Reduced Row Echelon Form (RREF). By identifying the pivot columns and free variables, the system expresses the solution vector in parametric vector form, effectively generating a basis for the null space.

Consider a 2 × 3 matrix A with rows [1, 2, -1] and [2, 4, -2]. To find its null space, we set up the homogeneous system Ax = 0. Notice that row 2 is a multiple of row 1, meaning we have one pivot variable (x₁) and two free variables (x₂ and x₃). Writing the equation in terms of free variables yields x₁ = -2x₂ + x₃. Expressing this in parametric vector form, the basis vectors for the null space span the solutions [-2, 1, 0]^T and [1, 0, 1]^T, capturing all possible linear combinations that satisfy the original matrix equation.

Double-check your matrix dimensions before inputting values to ensure the row and column counts align correctly with your linear system. Pay close attention to free variables during the interpretation phase, as they dictate the dimension of your null space (the nullity). Utilize reduced row echelon form steps to verify your manual homework calculations or engineering models efficiently.