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My understanding of a basis $B$ of a vector space $V$ is, that it is a set of linearly independent vectors that spans the vector space.

Therefore, the 0 vector is not included in the basis,

and proper subset of the basis cannot span V.

And here comes my question. I am not quite sure what it meas for "another basis for V to be disjoint from B".

I understand that a basis is not unique, but for two distinct bases to be disjoint, does it mean that the intersection of the two bases is empty, or their span are disjoint ?

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3 Answers 3

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It means exactly what it says: the bases themselves are disjoint as sets. For example, $\Bbb R^2$ has the disjoint bases $$B_1=\big\{\langle 0,1\rangle,\langle 1,0\rangle\big\}\quad\text{and}\quad B_2=\big\{\langle 1,1\rangle,\langle 1,-1\rangle\big\}\;:$$

$B_1\cap B_2=\varnothing$, so $B_1$ and $B_2$ are disjoint, and you can easily check that they span $\Bbb R^2$ and are linearly independent.

Note that it cannot mean that their spans are disjoint: if they are bases for $V$, each has $V$ as its span, so their spans are identical.

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Thank you very much for the detailed explanation with clear cut definitions. It helped a lot ! –  hyg17 Apr 11 '13 at 18:45
    
@hyg17: You’re welcome; glad it helped. –  Brian M. Scott Apr 11 '13 at 19:03
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It means the set of basis vectors are disjoint. The span of the two is not disjoint because the two bases span the same space.

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Thanks! That was helpful. –  hyg17 Apr 11 '13 at 18:45
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The span of any basis is the whole space. So the span of 2 bases cannot be disjoint. I guess, it is about that the 2 sets of the basis vectors are disjoint.

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Thank you for confirming my doubts ! –  hyg17 Apr 11 '13 at 18:46
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