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Math Help - four fundamental vector spaces

  1. #1
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    four fundamental vector spaces

    Compute the four fundamental vector spaces associated with A and verify the theorem "If A is a given matrix, then the null space is the orthogonal complement of the row space of A and the null space of A^{T} is the orthogonal complement of the column space of A."

    A= \begin{pmatrix}1 & 5 & 3 & 7\\2 & 0 & -4 & -6\\4 & 7 & -1 & 2\end{pmatrix}

    So far, I got rref(A)=B= \begin{pmatrix}1 & 0 & -2 & -3\\0 & 1 & 1 & 2\\0 & 0 & 0 & 0\end{pmatrix}

    I'm not sure if I am doing this write. The following is my work. Any advice or comments would be greatly appreciated. Thank you!

    x=\begin{pmatrix}2r+3s\\-r-s\\r\\s\end{pmatrix}=r\begin{pmatrix}2\\-1\\1\\0\end{pmatrix}+s\begin{pmatrix}3\\-2\\0\\1\end{pmatrix}

    S={\begin{pmatrix}2\\-1\\1\\0\end{pmatrix},\begin{pmatrix}3\\-2\\0\\1\end{pmatrix}}

    T={(1, 0, -2, -3), (0, 1, 1, 2)}

    and the transpose of A= \begin{pmatrix}1 & 2 & 4\\5 & 0 & 7\\3 & -4 & -1\\7 & -6 & 2\end{pmatrix}

    rref(transpose of A)= \begin{pmatrix}1 & 0 & {7/5}\\0 & 1 & {13/10}\\0 & 0 & 0\\0 & 0 & 0\end{pmatrix}

    S'=\begin{pmatrix}0\\-1\\1\\0\end{pmatrix}, \begin{pmatrix}-7/5\\-13/10\\0\\1\end{pmatrix}

    T'= \begin{pmatrix}1\\0\\7/5\end{pmatrix}, \begin{pmatrix}0\\1\\13/10\end{pmatrix}
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  2. #2
    Senior Member Twig's Avatar
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    Finding your vectors that Span Nul A looks correct.

    What about Col A?
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  3. #3
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    T'= Isn't this the basis for the column space of A?
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