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

determine whether W is a subspace of the vector space.
$$\displaystyle{W}={\left\lbrace{f}:{f{{\left({0}\right)}}}=-{1}\right\rbrace},{V}={C}{\left[-{1},{1}\right]}$$

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sonorous9n
See the vector space and checked wheather zero function belongs or not , if yes then check other conditions for subspace.
$$\displaystyle{W}={\left\lbrace{f}:{f{{\left({0}\right)}}}=-{1}\right\rbrace},{V}={C}{\left[-{1},{1}\right]}$$
∵ if $$\displaystyle{f}\in{W}\Rightarrow{f{{\left({0}\right)}}}=-{1}$$
and $$\displaystyle{0}{\left({0}\right)}={0}$$ [0 = zero function]
so $$\displaystyle{0}\notin{W}$$
so zero function does not belongs to W
so W is not subspace