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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

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1 Answer 1

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.)

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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

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