Is this statement true or false? Let U, V be non-zero subspaces of R3, and let W = U ∪ V be the set of vectors which lie in either U or V (or both). Then W is a subspace of R3 .
Asked
Active
Viewed 814 times
-1
-
can you share your thoughts/ attempts? – Siong Thye Goh Jan 30 '20 at 03:10
-
Here's the MathJax tutorial, and here are some general tips. Thanks for joining our community! – gen-ℤ ready to perish Jan 30 '20 at 03:14
-
I attempted to use the closures under vector addition and scalar multiplication to prove that W is a sub space of R. I haven’t learned about spanning sets yet. – statsontrack1122 Jan 30 '20 at 03:24
-
Does this answer your question? Union of two vector subspaces not a subspace? – mi.f.zh Jan 30 '20 at 03:28
-
Yes it does! Thank you! – statsontrack1122 Jan 30 '20 at 03:30
1 Answers
-1
Take $U$ to be the $x$ axis, and $V$ the $y$ axis. Then $(1,0,0) \in U$, $(0,1,0) \in V$, and clearly their sum is not in $W$.
You need more than the union in this case, since we have additional structure of linearity (i.e. a vector space) on this space. This is also (essentially) answered here.
This is also assuming you mean a linear/vector subspace, but I believe that's implicitly assumed in the question.
mi.f.zh
- 1,219