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Which are bases?

  1. Oct 20, 2011 #1
    1. The problem statement, all variables and given/known data

    Which of the following sets [itex]S[/itex] are bases for the vector space [itex]V=\mathbb{R}^3[/itex]?

    (a) [itex]S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}[/itex]

    (b) [itex]S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}[/itex]

    (c) [itex]S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} \right\}[/itex]

    (d) [itex]S=\left\{ \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} , \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} \right\}[/itex]

    3. The attempt at a solution

    By my reckoning, the only set that isn't a basis is (b) as it isn't linearly independent and [itex]\text{Span}(S)\neq V[/itex]. Correct?
     
  2. jcsd
  3. Oct 20, 2011 #2

    LCKurtz

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    Since there are only 3 and the dimension of R3 is 3, you are correct that whether or not they are linearly independent determines whether they are a spanning set (basis).
     
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