Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given two vector spaces $ V \subset W $ over a field $\mathbb{F}$ (where $V$ is a proper subspace of $W$ ). If we have three elements $x,y,z \in W$ . does the following two statements are true?(I can't find any reason for them to not be true, but it seems strange that both will be true)

(a) if $x,y,z$ are linearly independent as elements in $V$ , then they are also independent in $W$ .

(b) is $x,y,z$ are linearly independent as elements in $W$ , then they are also independent in $V$ .

What do you think? Is it true that both statements are correct?

Thanks in advance

share|cite|improve this question
up vote 3 down vote accepted

Yes, both are correct.

Both just say that if $\alpha x+\beta y+\gamma z=\vec 0$, where $\alpha,\beta,\gamma\in\Bbb F$, then $\alpha=\beta=\gamma=0_{\Bbb F}$. This works because the scalar multiplication, vector addition, and zero vector are the same in $V$ and $W$.

(And no, it’s not a stupid question: this is the sort of picky detail that you should worry about.)

share|cite|improve this answer
Great ! Thanks a lot ! (because of its picky nature, I decided to upload this question [which came up naturally during one of the theorems I encountered] Thanks again ! – joshua Sep 12 '12 at 16:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.