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) $(AB)^T=B^T A^T$

Author: mathtuition88

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