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Math Help - Basis for this set:

  1. #1
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    Basis for this set:

    S=\lbrace \begin{bmatrix} 0&0\\0&0 \end{bmatrix} \rbrace.

    I.e. it is a singleton.

    I've already showed that it is a subspace of M_2(R) (hopefully you'll agree that it is indeed a vector space).

    But the problem is:

    How can I find a basis for it?!

    A basis must span the set S, but it must also be a linearly independent set!

    So if I understand correctly, \lbrace \begin{bmatrix} 0&0\\0&0 \end{bmatrix} \rbrace spans S, but it is not linearly independent (since any singleton set containing just the 0 vector is linearly dependent).

    What am I to do?
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  2. #2
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    By definition, the span of the empty set is the zero vector.
    (kind of makes sense if we understand the concept of the empty sum
    evaluating to zero). Thus, your basis is: the empty set.

    Best,
    ZD
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  3. #3
    Super Member Gamma's Avatar
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    Iowa City, IA
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    Null set

    The null set is considered the basis for the unique 0 dimensional subspace of a vector space which consists of just the additive identity.
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  4. #4
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    Thanks guys! I knew it had to have something to do with that.
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