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Math Help - Writting Linear Functional as Direct Sum.

  1. #1
    Junior Member qspeechc's Avatar
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    Linear Functionals and Direct Sums.

    Hi everyone

    Sorry about the goofy thread title, it should read "Linear Functionals and Direct Sums".

    Let V be a finite-dimensional vector space over the field of scalars F, and T a linear map from V to F. Show that
    V = kerT\oplus span\{ u\}
    where Tu \neq 0

    Well, I reasoned that if T is not the zero map, then the range of T is F (simple to show), in which case the range of T is spanned by one vector. From dim(V)=dim(kerT)+dim(ranT) we must have that the dimension of the kernel is one less than that of V. So let \{u_1,\ldots ,u_k \} be a basis for kerT. Extending it to a basis to V we only need add u, which is not in the kernel, because if it was it would contradict the linear independence of the base for the kernel.
    So a basis for V is \{u_1,\ldots ,u_k, u \}. Obviously kerT \cap span\{ u\} = 0.
    The fact that \{u_1,\ldots ,u_k, u \} is a basis for V shows that V = kerT + span\{ u\}. Thus, V = kerT\oplus span\{ u\}.
    If T is the zero map then the set of vectors not in the kerel is the empty set, and so span\{ u\}=\{ 0\}, and everything is all good.
    Is this right?
    Last edited by qspeechc; February 28th 2010 at 02:26 AM.
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