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Thread: Linear Algebra; prove the theorems on left and right inverses.

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    Linear Algebra; prove the theorems on left and right inverses.

    Prove that if the matrix $\displaystyle A$ has a right inverse, then for each $\displaystyle b$
    the equation $\displaystyle Ax = b$ has at least one solution. If $\displaystyle A$ has a left inverse,
    then that equation has at most one solution.
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  2. #2
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    Quote Originally Posted by mathwizard View Post
    Prove that if the matrix $\displaystyle A$ has a right inverse, then for each $\displaystyle b$
    the equation $\displaystyle Ax = b$ has at least one solution.
    let $\displaystyle AB=I,$ with $\displaystyle I$ identity matrix. then $\displaystyle ABb=Ib=b,$ i.e. $\displaystyle x=Bb$ satisfies $\displaystyle Ax=b.$ so there's at least one solution.

    If $\displaystyle A$ has a left inverse, then that equation has at most one solution.
    let $\displaystyle CA=I,$ and suppose that $\displaystyle Ax=Ay=b.$ then $\displaystyle x=Ix=CAx=CAy=Iy=y.$ so $\displaystyle Ax=b$ has at most one solution.
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