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Math Help - matrix proving problem

  1. #1
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    Unhappy matrix proving problem

    Suppose u and v are solutions to the linear system Ax=b. Show that if scalars α and β satisfy α+β=1, then αu+βv is also a solution to the linear system Ax=b.
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  2. #2
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    Quote Originally Posted by mikai View Post
    Suppose u and v are solutions to the linear system Ax=b. Show that if scalars α and β satisfy α+β=1, then αu+βv is also a solution to the linear system Ax=b.
    Just plug \alpha u+\beta v for x into the left side of Ax=b and use the linearity of applying A to that vector to simplify everything to just b

    Ax = A(\alpha u+\beta v)=\alpha (Au)+\beta (Av)=\ldots = b
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