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Simple fundamental subspaces problem stumping me

  1. Dec 10, 2015 #1
    So the matrix is just a row vector

    [tex]\begin{bmatrix}3 & 4 & 0\end{bmatrix}[/tex]

    My problem, is that I get the nullspace as having to 2 dimensions, and the row space as having 2 dimenions, but that adds up to 4 dimensions, when it should add up to three. What simple thing am I missing?

    Null space base vectors

    [tex]\begin{bmatrix}-4\\3\\0\end{bmatrix},\begin{bmatrix}0\\0\\1\end{bmatrix}[/tex]

    Row space base vectors

    [tex]\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}[/tex]
     
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  3. Dec 10, 2015 #2

    HallsofIvy

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    Taking the matrix to be the single row [itex]\begin{bmatrix}3 & 4 & 0\end{bmatrix}[/itex] then the null space is all [itex]\begin{bmatrix}x \\ y \\ z \end{bmatrix}[/itex] such that [itex]\begin{bmatrix}3 & 4 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}= 3x+ 4y= 0[/itex]. Yes, that is of dimension 2 having [itex]\{\begin{bmatrix}-4 \\ 3 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}\}[/itex] as basis. However, the row space is one dimensional, having the vector [itex]\begin{bmatrix}3 & 4 & 0 \end{bmatrix}[/itex], the single row making up the matrix, as basis.
     
  4. Dec 10, 2015 #3

    Geofleur

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    Those two last vectors cannot be in the row space, because a linear combination of them will give the first null space vector, ## [-4 \ 3 \ 0]^T##. In fact, the row space is the space spanned by the rows of the matrix, considered as vectors. So the row space consists of all multiples of ## [3 \ 4 \ 0]^T ##.

    EDIT: oops, HallsofIvy beat me to it :-)
     
  5. Dec 10, 2015 #4
    I realize what I was missing now.

    I kept thinking of the row space as all those vector that can be operated on by the matrix without giving a zero vector result. Except that's not what the rowsapce is! It's the vector/s that characterize/s the rows of the matrix. Thanks @HallsofIvy and @Geofleur!
     
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