 Let W be a subset of the vector space V Frank Guyton 2022-01-06 Answered
Let W be a subset of the vector space V where u and v are vectors in W. If ($$\displaystyle{u}\oplus{v}$$) belongs to W, then W is a subspace of V:
Select one: True or False

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Subspace: Suppose that V is a vector space and W is a subset of $$\displaystyle{V},{W}\subseteq{V}$$. Endow W with the same operations as V. Then W is a subspace if and only if three conditions are met
- W is non-empty, $$\displaystyle{W}\ne\emptyset$$.
- If $$\displaystyle{x}\in{W}$$ and $$\displaystyle{y}\in{W}$$, then $$\displaystyle{x}+{y}\in{W}$$.
- If $$\displaystyle\alpha\in{R}$$ and $$\displaystyle{x}\in{W}$$, then $$\displaystyle\alpha{x}\in{W}$$.
The given statement is, '' Let W be a subset of the vector space V where u and v are vectors in W. If (u+v) belongs to W, then W is a subspace of V.'' Since the remaining two conditions are not satisfied, therefore the given statement is False.