Thread: dimension

Let V be a finite dimensional vector space. Show that if W1,....,Wn are subspaces of V such that none of these subspaces are qeual to V, then Union of all these subspaces does not equal V.

i am thinking induction on dim V might help but do not really have idea how to go bout it. so please help.

Originally Posted by Kat-M

Let V be a finite dimensional vector space. Show that if W1,....,Wn are subspaces of V such that none of these subspaces are qeual to V, then Union of all these subspaces does not equal V.

The union of two subpaces is a subspace if and only one is contained in another.
Thus if is a subspace thus and .
And so .

Originally Posted by ThePerfectHacker

The union of two subpaces is a subspace if and only one is contained in another.

The reason that is so: Suppose u is in subspace but NOT subspace and v is in subspace but not in . The u+ v cannot be in . If it were, then it would have to be in either or (or both). If u+ v were in , then, because is closed under addition and scalar multiplication u+ v+ (-u)= v would be in , a cotradiction. If u+ v were in , simlarly u+ v+ (-v)= u would be in , again a contradiction.