Superposition Principle of Linear Algebra

Superposition Principle

If \mathbf{u} is a solution to \bf{A}\mathbf{x}=\mathbf{b}, and \mathbf{v} is a solution to \bf{A}\mathbf{x}=\mathbf{c}, then \mathbf{u}+\mathbf{v} is a solution to \bf{A}\mathbf{x}=\mathbf{b}+\mathbf{c}.

Proof:
\mathbf{A}(\mathbf{u}+\mathbf{v})=\mathbf{A}\mathbf{u}+\bf{A}\mathbf{v}=\mathbf{b}+\mathbf{c}

Properties of Matrix Transpose

1) (A^T)^T=A
2) (A+B)^T=A^T+B^T
3) (kA)^T=kA^T
4) (AB)^T=B^T A^T

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