Doubt on subspace of a vector space

Question

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

Answer 1

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 $0x=\vec 0\in S$.

Answer 2

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

Answer 3

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.