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I'm trying to show that the functions $c_1 + c_2 \sin^2 x + c_3 \cos^2 x$ forms a vector space.

And I will need to find a basis of it, and its dimension.

Is there a way how to do this without verifying the 8 axioms for a vector space, and if we let the set $X = \{c_1 + c_2 \sin^2 x + c_3 \cos^2 x\}$ then we note that $1 = \sin^2 x + \cos^2 x$, and this is enough. So the dimension is $2$. Thanks.

Can you please provide clarification on how the argument of the subspace of the vector space follows? I think you did it already by inspection, but its not very complete to me, can you please write it down? Thanks

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    If by the above you meant "the span of..." then it is trivial: the span of *anything* within a vector space, including the empty set, is **always** a vector space.2012-08-25
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    I'm always amazed how undergraduate algebra can "cram four axioms into eight axioms". Sort of like baseball "crams two minutes of action into two hours".2012-08-26
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    a bounty? Really? If you'd like more help, why don't you show us how far you've gotten, and where you're getting stuck?2012-08-30

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