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Math Help - span

  1. #1
    Senior Member Sampras's Avatar
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    span

    1. Find three distinct non-zero vectors  u,v,w in  \mathbb{R}^{3} such that  \text{span}( \{u,v \}) = \text{span}( \{v,w \}) = \text{span}( \{u,v,w \}) but such that  \text{span}(\{u,w \}) \neq \text{span}(\{u,v,w \}) .

    Consider  u = (1,0,0) ,  v = (0,1,0) , and  w = (1/2,0,0) . Then  \text{span}( \{u,v \}) = \text{span}( \{v,w \}) = \text{span}( \{u,v,w \}) . But  \text{span}(\{u,w \}) \neq \text{span}(\{u,v,w \}) .

    2. Find a basis for  M_{2 \times 2}^{0} (\mathbb{R}) , the vector space of  2 \times 2 matrices with trace zero. Why is this set a basis?

    So we want to find a spanning set that is linearly independent. Consider  B = \left \{ \begin{bmatrix} 1 & a \\ b & -1 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right \} where  a and  b can be chosen freely. This is a basis because it is a linearly independent set that spans  M_{2 \times 2}^{0}(\mathbb{R}) .

    Is this correct?
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  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
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    Iowa City, IA
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    517
    Part 1 looks good to me, but for number two, your second basis element is not even in the set Tr(I_2)=1+1=2 \not = 0.
    I think these are the ones you want.
    \left[\begin{array}{cc}1 & 0 \\0 & -1\end{array}\right]
    \left[\begin{array}{cc}0 & 1 \\0 & 0\end{array}\right]
    \left[\begin{array}{cc}0 & 0 \\1 & 0\end{array}\right]

    Linear independence and spanning are both pretty clear.
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