how to use the basis of a nullspace to quickly determine whe

how to use the basis of a nullspace to quickly determine whether a solution to Ax=b exists for any given b

Solution

Let A = [A1|A2| · · · |An] be an m × n matrix, with columns A1, A2, . . . , An.

Then a solution x = x1 x2 . . . xn

to the matrix equation Ax = b satisfies x1A1 + x2A2 + · · · + xnAn = b.

The equation Ax = b has a solution exactly when b is a linear combination of the columns of A.

• The column space is used to decide whether (4.4) is solvable or not. This equation is solvable if and only if b R(A).

• The null space is used to decide how many solutions you have, once you know that you have at least one, i.e. b R(A). If the nullspace is trivial, N(A) = {0}, then there is exactly one solution. If the nullspace is nontrivial, then you have free parameters in the general solution to (4.4). The number of free parameters equals to the nullity of A.

• The set of solutions to Ax = b is an affine set (“affine subspace”). It has the form x = x0 + N(A) = {x0 + y : y N(A)} where x0 is any fixed solution. In other words, the solution set is an affine set which is a translate of N(A). The translating vector x0 is certainly not unique.