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zapz
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Homework Statement
This is a problem from chapter 1.3 of Linear Algebra by F/I/S.
Let [itex]W_{1}[/itex] and [itex]W_{2}[/itex] be subspaces of a vector space V. Prove that [itex]W_{1}[/itex] [itex]\cup[/itex] [itex]W_{2}[/itex] is a subspace of V iff [itex]W_{1}[/itex][itex]\subseteq[/itex][itex]W_{2}[/itex] or [itex]W_{2}[/itex] [itex]\subseteq[/itex] [itex]W_{1}[/itex].
Homework Equations
See attempt at solution.
The Attempt at a Solution
My proof goes as such:
If [itex]W_{1}[/itex][itex]\subseteq[/itex][itex]W_{2}[/itex] then the union of those subspaces is [itex]W_{2}[/itex], therefore, by the given, it the union is a subspace of V.
The same logic is used to argue the other subset.
I'm not sure if this is correct, and additionally, I'm not sure if its a logical proof. I feel it is a little cyclical maybe. Thanks for all the help, I'm having a tough time with this text.