# A question regarding the basis of a Vector Space.

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 ?

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.

• 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

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.

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.

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