91Ó°ÊÓ

The null spaceof matrix A is the set Nul Aof all solutions of the homogeneous equation \(Ax = 0\).

Theorem 5 states that A is an invertible \(n \times n\) matrix; then for each b in \({\mathbb{R}^n}\), equation \(A{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) has the unique solution \({\mathop{\rm x}\nolimits} = {A^{ - 1}}{\mathop{\rm b}\nolimits} \).

Suppose \(P\) is a \(5 \times 5\) matrix and \({\mathop{\rm Nul}\nolimits} \,\,P\)is the zero subspace.

When \({\mathop{\rm Nul}\nolimits} \,\,P = \left\{ 0 \right\}\)the equation \(P{\mathop{\rm x}\nolimits} = 0\) has only a trivial solution. The Invertible Matrix theorem states that \(P\) is invertible and equation \(P{\mathop{\rm x}\nolimits} = {\mathop{\rm b}\nolimits} \) has a solution for each b in \({\mathbb{R}^5}\). In addition, each solution is unique according to theorem 5.

Thus, equation \(Px = {\mathop{\rm b}\nolimits} \) has a unique solution for each b in \({\mathbb{R}^5}\).