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Math Help - Image of L is contained in M

  1. #1
    Senior Member sfspitfire23's Avatar
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    Image of L is contained in M

    Let A be a 4x3 matrix and let B be a 4x4 matrix. BA is then a 4x3 matrix. Let L be a linear transformation from R^4 to R^3 defined by L(x) = xA where x is a vector. Let M be the linear transformation from R^4 to R^3 defined by M(x) = xBA.
    Show that the image of M is contained in the image of L.

    What I have-

    Take any vector x. The image of L and M will be the span of the rows of A and the rows of BA, or the set of all linear combinations. So, we have:

    [x1 x2 x3 x4]*A = x1A1 + x2A2 + x3A3 + x4A4

    and

    [x1 x2 x3 x4]*BA = x1BA1 + x2BA2 + x3BA3 + x4BA4

    I'm stuck at this point. I think I have to invoke the linearity fact and add these together somehow but i'm not seeing itů

    Thanks.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Image of L is contained in M

    It is more simple: you needn't the concept of linear combination, only the definition of image. If y\in\mbox{Im }M, then there exists x\in\mathbb{R}^4 such that y=xBA=(xB)A. As a consequence, y\in\mbox{Im }L.
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