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).(adsbygoogle = window.adsbygoogle || []).push({});

The first question says:

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

And the answer is:

Take [itex]U_{1}=\{(x_{1}, x_{2})\in\Re^{2}:x_{2}=0\}[/itex] and [itex]U_{2}=\{(x_{1}, x_{2})\in\Re^{2}:x_{1}=0\}[/itex]

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 [itex](x_{1}, x_{2})[/itex] in [itex]\Re^{2}[/itex] such that [itex]x_{1}=1[/itex]. What is the span of S?

And the answer is:

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

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

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

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# Question on subspaces and spans of vector spaces

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