# Thread:

1. ## 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.

2. 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 $W_1\cup W_2$ is a subspace thus $W_1\subseteq W_2$ and $W_2\subseteq W_1$.
And so $\dim (W_1\cap W_2) = \max \{ \dim (W_1),\dim (W_2) \} < \dim (V)$.

3. 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 $U_1$ but NOT subspace $U_2$ and v is in subspace $U_2$ but not in $U_1$. The u+ v cannot be in $U_1\cup U_2$. If it were, then it would have to be in either $U_1$ or $U_2$ (or both). If u+ v were in $U_1$, then, because $U_1$ is closed under addition and scalar multiplication u+ v+ (-u)= v would be in $U_1$, a cotradiction. If u+ v were in $U_2$, simlarly u+ v+ (-v)= u would be in $U_2$, again a contradiction.
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