# Question on subspaces and spans of vector spaces

Hi, I have read my notes and understand the theory, but I am having trouble understanding the following questions which are already solved (I am giving the answers as well).

The first question says:

Let $U_{1}$ and $U_{2}$ be subspaces of a vector space V. Give an example (say in $V=\Re^{2}$) to show that the union $U_{1}\bigcup U_{2}$ need not be a subspace of V.

Take $U_{1}=\{(x_{1}, x_{2})\in\Re^{2}:x_{2}=0\}$ and $U_{2}=\{(x_{1}, x_{2})\in\Re^{2}:x_{1}=0\}$

So I really don't understand this answer... I would really appreciate it if someone could explain it to me.

And the second question says:

Let S be the set of all vectors $(x_{1}, x_{2})$ in $\Re^{2}$ such that $x_{1}=1$. What is the span of S?

span S = $\Re^{2}$ because $(x_{1}, x_{2})=x_{1}(1, x^{-1}_{1}x_{2})$ when $x_{1}\neq 0$ and $(x_{1}, x_{2})=(1, 0)-(1, -x_{2})$ when $x_{1}=0$.

So I don't understand this explanation, mainly because I thought that $x_{1}$ is supposed to be 1... :/

Sorry if the questions sound silly, and thanks for any help you can give me! :)

For the first one: what don't you understand?? The solution claims three things:

1) U1 is a subspace of $\mathbb{R}^2$
2) U2 is a subspace of $\mathbb{R}^2$
3) $U_1\cup U_2$ is not a subspace of $\mathbb{R}^2$

Which of these three points is bothering you??

For the second one: A typical vector in S is (1,a) for a arbitrary. We are interesting in whether the span of S is $\mathbb{R}^2$.
The span is all linear combinations of elements of S. Thus if we claim that the span of S is $\mathbb{R}^2$, then we actually claim that each vector in $\mathbb{R}^2$ is a linear combination of elements in S. Can you find such a linear combination?

Fredrik
Staff Emeritus