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Under following conditions

$ a, b \in \mathbb{R}, V = \mathbb{R}^{2}, W = \{(x, y)\ |\ ax + by = 0 \} $

is W a subspace of V? I know the basics, but how would I prove that addition and multiplication are closed over this subset?

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1 Answer 1

up vote 6 down vote accepted

Let $(v_1,v_2),(w_1,w_2)\in W.$ We have $$\begin{cases} av_1+bv_2=0\\ aw_1+bw_2=0 \end{cases} $$ What happens when you add these equations? What happens when you multiply the first equation by a real number $r$?

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Thanks, this is actually very clear to me now.. I feel stupid for not seeing that myself. So it is obviously a subspace, since both new equations you mention are still zero? –  Mats Jan 24 '12 at 23:28
    
Comment corrected. Yes. Only you need to use the algebraic properties of the real numbers to see the vectors $(v_1+w_1,v_2+w_2)$ and $(rv_1,rv_2)$ in the resulting equations. –  user23211 Jan 24 '12 at 23:39
    
Yes, of course. Thanks a lot! –  Mats Jan 24 '12 at 23:49

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