What are disjoint vector spaces?

Question

I want to clarify what does represent disjoint vector space? I know that this terminology related to set is very easy, because disjoint set is for example given two set

$A={1,2,3}$

and

$B={4,5,6}$

or in another words, two set which has not any common element is called disjoint set, but for vector space what does it means? In my book there is written such kind of sentence

If $V_1$ and $V_2$ are essentially disjoint vector spaces (not just spaces of vectors), the sum is called the direct sum. This relation is denoted by

enter image description here

as I know this symbol denotes tensor or Kronecker product, I have idea that disjoint vector spaces should have also disjoint vector subspace,which means that coefficient of linear combinations,by which vector space can be generated from it's vector subspace,must be always disjoint,or set of coefficients more correctly.is it like this or it denotes different one? Thanks

$\otimes$ is tensor product whereas $\oplus$ is just direct sum. — blitzer, May 01 '13 at 08:04

score 2 · Accepted Answer · answered May 01 '13 at 08:11

2

The writing

enter image description here

means that $V_1$ and $V_2$ are subspaces of $V$, such that every vector $v$ in $V$ can be written, in one and only one way, as a sum $$v=v_1+v_2$$ where $v_1\in V_1$ and $v_2\in V_2$. The unicity part of this statement is equivalent to the fact that $V_1$ and $V_2$ trivially intersect $$V_1\cap V_2=\{\vec 0\}$$ Another way to characterize the direct sum of two subspaces could be that $V_1\oplus V_2$ is the subspace generated by the union $V_1$ and $V_2$, meaning the smallest subspace of $V$ containing both vectors of $V_1$ and $V_2$

answered May 01 '13 at 08:11

Federica Maggioni

8,422

thanks for reply,now one simple question just what i have,intersection of this subspace,does this intersection is the same meaning as for example intersection of two set?so it means that this two subspace are like set and they does not contain common element? – dato datuashvili May 01 '13 at 08:18
@dato Yes, the motivation is the following: as for other structures(e.g. ideals in a ring, sub-modules in a module) the insiemistic intersection preserves the algebraic structure, but the union not, hence the intersection of two vector spaces is again a vector space, while for the union you have to take a "closure", in this case a linear closure, which is actually the direct sum – Federica Maggioni May 01 '13 at 08:23
@dato ...but remember that a vector subspace of $V$ ALWAYS contains the zero-vector, since a vector subspace is in particular an additive subgroup of $(V,+)$ hence it must contain the neutral element w.r.t addiction, which is the zero vector – Federica Maggioni May 01 '13 at 08:25
thanks ones again ,i found following question http://math.stackexchange.com/questions/207120/disjoint-union-of-subsets-and-direct-sum-of-subspaces-clarify-explanation so it means that two vector space is disjoint only if their intersection is $0$ vector,{0},because in this case any linear combination from this vector is useless and it will stay {0} zero vector again.thanks very much – dato datuashvili May 01 '13 at 08:28

score 1 · Answer 2 · answered May 01 '13 at 08:08

1

The easiest way to think of disjoint vector spaces is that $V_1$ and $V_2$ are disjoint if

$V_1 \cap V_2 = \{ \vec{0} \}$

answered May 01 '13 at 08:08

blitzer

458
2
3

but what if they contain element zero? – dato datuashvili May 01 '13 at 08:09
i want to clarify that is it same as for set? – dato datuashvili May 01 '13 at 08:11
I assume your book means they are subspaces of a larger vector space. If you have two vector spaces which aren't compatible, you could think of them as being disjoint as sets, in which case their intersection would be empty – blitzer May 01 '13 at 08:13

What are disjoint vector spaces?

2 Answers2