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$V$ is a vector space. $u,v,w \in V$ and $u+2v+3w=0$. Show that $\mathrm{Span}(u,v)=\mathrm{Span}(v,w)$ or not?

I really struggled with this question. Can anybody help me?

Glorfindel
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usereb
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1 Answers1

3

Hints:

Assuming that char$\,F\neq3\;,\;\;F:=$ the field over which the linear space is defined , check that

$$w=-\frac13u-\frac23v\;\;,\;\;\text{and also}\;\;u=-2v-3w$$

If char$\,F=3\;$ we have a counter example with $\;u=v=0\;,\;\;w\neq0\;$:

DonAntonio
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  • Span(u,v) is in Span(v,w) always, but we cannot say anything about the other inclusion if the field has characteristic 3. Pointing it out for all viewers, because I missed it at first glance. – Guy May 27 '17 at 10:02
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    @Sabyasachi Good point, thanks. Anyway, this question is so basic that most probably the OP's working with $\;\Bbb R\;,\;\;\Bbb C\;$ or something similar. But I shall edit the answer. – DonAntonio May 27 '17 at 10:04
  • @Guy Thanks. I have a problem because I can't use characteristic. This problem in our linear algebra lesson and characteristic is not in our topics. If I use this it won't be accepted. I should build a field that has characteristic 3 – usereb May 28 '17 at 07:31
  • @Guy I can't think any other set for giving example except /BbbZ_3 – usereb May 28 '17 at 07:37
  • @esrabasar If you don't have characteristic in your topics, then just ignore whatever anybody has said here about characteristic, – Guy May 28 '17 at 07:37
  • @Guy It is really hard to find a counter exp without characteristic for me – usereb May 28 '17 at 07:40
  • @Guy thank you so much I hope it will be accepted – usereb May 28 '17 at 07:42
  • @esrabasar i made a correction in my comment, check. $F_p(X) = \{f/g| f,g \in \mathbb{Z}_3[X], g\ne 0\}$ is an example of an infinite field with characteristic 3if you need one. Check this [answer](https://math.stackexchange.com/questions/58424/example-of-infinite-field-of-characteristic-p-neq-0#58425) also – Guy May 28 '17 at 07:45
  • @Guy Thanks again I have realised that \mathbb{Z}_3[x] is an integral domain. f/g is very good idea :) – usereb May 28 '17 at 10:33