• $$\vec{u}=\underset{\sim}{A}\vec{v}$$, or $$\underset{\sim}{A}\vec{v}=\vec{u}$$, represents a set of linear equations that can (usually) be solved for $$\vec{v}$$. If the matrix is square and $$\operatorname{det}(\underset{\sim}{A})\neq 0$$, then $$\vec{v}=\underset{\sim}{A}^{-1}\vec{u}$$.
• A homogeneous set of equations has the form $$\underset{\sim}{A}\vec{v}=0$$, i.e., it has $$\vec{v}=0$$ as a solution (like homogeneous differential equations). In this case, nonzero solutions for $$\vec{v}$$ exist only if $$\operatorname{det}(\underset{\sim}{A})=0$$.