2.3: Systems of linear equations
- Page ID
- 18055
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• \(\vec{u}=\underset{\sim}{A}\vec{v}\), 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 \(\operatorname{det}(\underset{\sim}{A})=0\).