Exercise · ZW-VECTOR-SUBSPACE-009 · ★★★ Advanced
Let \(U\) be the \(x\)-axis and \(W\) the \(y\)-axis in \(\mathbb{R}^2\). Why is \(U\cup W\) not a subspace ?
Choose one option.
A It does not contain the zero vector B It is not closed under scalar multiplication C \((1,0)\) and \((0,1)\) are in the union, but their sum \((1,1)\) is not D It is: a union of subspaces is always a subspace Show answer C. \((1,0)\) and \((0,1)\) are in the union, but their sum \((1,1)\) is not← Back to the question
Explanation Each axis is a subspace and the union contains \(0\) and is closed under scalars, but addition fails. The intersection, in contrast, is always a subspace .
A vector subspace \(W\) of a vector space \(V\) is a nonempty subset that is itself a vector space under the same operations.
\[0\in W,\quad u,v\in W\Rightarrow u+v\in W,\quad c\in\mathbb{R},\ v\in W\Rightarrow cv\in W\]
More practice problems: vector subspace