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.

- 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.