# determine whether W is a subspace of the vector space. W

determine whether W is a subspace of the vector space.
$$\displaystyle{W}={\left\lbrace{\left({x},{y}\right)}:{x}-{y}={1}\right\rbrace},{V}={R}^{{2}}$$

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Paul Mitchell
By the definition of a subspace a nonempty subset W of a vector space is called subspace of V, if W is a vector space under the operations of addition and scalar multiplication defined in V.
From the above definition it is clear that a subspace must also be a vector space itself and thus must satisfy the axioms of a vector space.
However, the additive identity for $$\displaystyle{R}^{{2}}$$ that is,(0,0) is not in the subset W .
Hence, all the vector space axiom are not satisfied by W, so it is not a vector space, and hence not a subspace.
Therefore, W is not a subspace of $$\displaystyle{R}^{{2}}$$.