(Solved): H3.4: Superposition principle for matrix equations Suppose we know \( \underline{\mathbf{x}}_{i} \ ...

H3.4: Superposition principle for matrix equations Suppose we know \( \underline{\mathbf{x}}_{i} \) such that \( \underline{\underline{\mathbf{M}}} \underline{\mathbf{x}}_{i}=\underline{\mathbf{f}}_{i} \) for two or more problems \( (i=1,2, \ldots, N) \). Suppose we wish to solve the problem \( \underline{\underline{\mathbf{M}}} \underline{\mathbf{x}}=\underline{\mathbf{f}} \) where \( \underline{\mathbf{f}}=\sum_{i} \alpha_{i} \underline{\mathbf{f}}_{i} . \) Show that \( \underline{\mathbf{x}}=\sum_{i} \alpha_{i} \underline{\mathbf{x}}_{i} \) is a solution.