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I am new to linear algebra and I have a doubt that : in 2D coordinate system is a line which is at 45 degree NOT passing through the origin a subspace of the vector space comprising the whole 2D plane i.e. $ \mathbb{R}^2 $ ? let $V = \{ (x,y) \in \mathbb{R}^2 \}$. and $W = \{ (x,y) \in \mathbb{R}^2 : y-x+1=0 \}$ so is $W$ a subspace of $V$...please correct me if my expressions to express vector spaces/sub spaces are wrong...thank you

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Thanx guys for clearing my doubt...a +1 to all coz I cant accept ony one's answer as correct as all are correct... –  rotating_image Feb 13 '13 at 1:34

3 Answers 3

A subspace $S$ of a vector space must always contain the zero vector, because (it is not allowed to be the empty set, and) (i) it must be closed under subtraction: if $x\in S$ then $x-x=\vec 0\in S$, or alternatively (ii) it must be closed under scalar multiplication if $x\in S$ then $0x=\vec 0\in S$.

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No, subspaces alway contain the null vector, since if they contain $(x,y)$, they contain $(ax,ay)$ for any real $a$, including $a=0$.

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Your $W$ is not a linear subspace, which is what we mean when we say "subspace" in a linear algebra course. But $W$ is an affine subspace, a term used in mathematics as well.

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Can I know some few words about affine subspace?or if you give me some reference...I have gone through wiki and couldnt get the phrase "affine properties of euclidian space" –  rotating_image Feb 13 '13 at 1:45
    
@rotat: Maybe search here, and not general Google... for example math.stackexchange.com/questions/185768 ... A nice textbook on geometry (including affine geometry) is Coxeter Introduction to Geometry. –  GEdgar Feb 13 '13 at 15:04

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