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 April 25th, 2010, 12:48 PM #1 Senior Member   Joined: Dec 2006 Posts: 167 Thanks: 3 A infinite dimensional linear space Let $B= \left\{ \sqrt{x} \; | \; x = 1\text{ or } x\text{ is prime}\right\},\; S = \left\{ \sum_{k=1}^n a_k b_k \;|\; n\in N, \;a_k \in Q,\; b_k \in B\\right}$ Prove that $S$ is a linear space over $Q$ with a basis $B$
 April 25th, 2010, 01:23 PM #2 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: A infinite dimensional linear space What rules must S follow to be a linear space over Q?
 April 25th, 2010, 03:12 PM #3 Senior Member   Joined: Dec 2006 Posts: 167 Thanks: 3 Re: A infinite dimensional linear space $a,b \in S, \; s,t \in Q \Rightarrow sa+tb \in S$
 April 26th, 2010, 08:05 AM #4 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: A infinite dimensional linear space Is that really all? (I thought there was more... but I may be wrong) Anyway, does S satisfy this? If so, what's left is to show that B is a basis. This should be immediate. THe definition of S gives us this directly.
 April 29th, 2010, 10:52 AM #5 Senior Member   Joined: Dec 2006 Posts: 167 Thanks: 3 Re: A infinite dimensional linear space You have to prove that $\forall m \in N^+ \; \forall b_1,\cdots, b_m \in B \; ((x_1b_1+ \cdots + x_m b_m= 0\quad \quad x_k \in Q) \Rightarrow (x_k = 0\quad k=1,\cdots,m))$ That is, to prove B is linear independent over Q

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