Thread: dimension
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Old January 12th, 2009, 07:42 AM
HallsofIvy HallsofIvy is offline
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Originally Posted by ThePerfectHacker View Post
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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