vector space over an infinite field is not a finite union of proper subspaces
Let where each is a proper subspace of and is minimal. Because is minimal, .
Let and let .
Define . Since is not the zero vector and the field is infinite, must be infinite.
Since one of the must contain infinitely many vectors in .
However, if were to contain a vector, other than , from there would exist non-zero such that . But then and we would have contrary to the choice of . Thus cannot contain infinitely many elements in .
If some contained two distinct vectors in , then there would exist distinct such that . But then and we would have contrary to the choice of . Thus for cannot contain infinitely many elements in either. ∎
|Title||vector space over an infinite field is not a finite union of proper subspaces|
|Date of creation||2013-03-22 17:29:43|
|Last modified on||2013-03-22 17:29:43|
|Last modified by||loner (106)|