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
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
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$\begingroup$The writing
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 $V_2$
$\endgroup$ 4 $\begingroup$The easiest way to think of disjoint vector spaces is that $V_1$ and $V_2$ are disjoint if
$V_1 \cap V_2 = \{ \vec{0} \}$
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