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N(A) and R(A) in terms of their basis

  1. Oct 6, 2011 #1

    sharks

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    Gold Member

    1. The problem statement, all variables and given/known data
    The matrix A =
    1 1 1 1
    -1 0 1 0
    1 2 3 2

    Express null space and row space of A in terms of their basis vectors.

    2. The attempt at a solution

    I have found the null space to be: x3 [1 -2 1 0]^T + x4 [0 -1 0 1]^T.

    But my problem is how do i write the final answer correctly? Should i just write the answer as above? Or should i just write it this way: [1 -2 1 0]^T and [0 -1 0 1]^T

    I did a search online and ended up with this way to present the solution, but there are so many variations, i'm confused.
    {[1 -2 1 0]^T, [0 -1 0 1]^T}.

    Which is the correct established answer format?

    For the row space, i gave the answer like this:
    [ 1 1 1 1] and [0 1 2 1]

    Or should it be like this?: {[ 1 1 1 1], [0 1 2 1]}
     
    Last edited: Oct 6, 2011
  2. jcsd
  3. Oct 6, 2011 #2

    HallsofIvy

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    The problem says to "Express the null space of A in terms of its basis vectors.
    So I would say you can give the answer in either of two ways:
    1) "All vectors in the null space of A are of the form [itex]x_1\begin{bmatrix}1 \\ -2 \\ 1 \\ 0\end{bmatrix}+ x_2\begin{bmatrix}0 \\ -1 \\ 0 \\ 1\end{bmatrix}[/itex]"
    or
    2) "A basis for the null space is [itex]\{\begin{bmatrix}1 \\ -2\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix}0 \\ -1 \\ 0 \\ 1\end{bmatrix}\}[/itex]".

    But the words explaining what you answer means are as important as your vectors.
     
  4. Oct 6, 2011 #3

    sharks

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    Thanks for your help, HallsofIvy. I will keep your advice in mind for my exams.
     
  5. Oct 6, 2011 #4

    HallsofIvy

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    Yep- actually writing out full sentence answers is likely to send your teacher into shock!
     
  6. Oct 6, 2011 #5
    You can also say that the null space is span{(v1), (v2)}.
     
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