Answer to Question #277349 in Linear Algebra for Dani

Question

Give an example showing that the union of two subspace of a vector space V over a filed F is not necessarily a subspace of V over F.

Answer 1

$\text{The reason why this can happen is that all vector spaces, and hence subspaces too, }\\\text{must be closed under addition (and scalar multiplication). The union of two subspaces}\\\text{takes all the elements already in those spaces, and nothing more. In the union of}\\\text{ subspaces $W_1$ and $W_2$ there are new combinations of vectors we can add together that}\\\text{we couldn't before, like like $v_1+w_2$ where v1$\in$ W1 and w2∈W2. }\\ \text{For example, take $W_1$ to be the x-axis and $W_2$ the y-axis, both subspaces of $\mathbb{R}^2$ .}\\ \text{Their union includes both (3,0) and (0,5) whose sum (3,5) is not in the union. Hence }\\ \text{the union is not a vector space}$