Let A be an n times m matrix over k and consider the homogen

Let A be an n times m matrix over k and consider the homogeneous system of linear equations xA = 0 where x k^n. We know that x = 0 is a solution; what else can we say about the solution space s = {x K^n | xA = 0}? (6) Let W denote the row space of A, and show how we can con-sider x s to be an element of the annihilator of W. Now show that S = A(W).

Solution

(5)   Every homogeneous system has at least one solution, known as the zero solution (or trivial solution), which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix (det(A) 0) then it is also the only solution.

If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties:

These are exactly the properties required for the solution set to be a linear subspace of Kⁿ.

In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A. Numerical solutions to a homogeneous system can be found with an SVD decomposition.