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
Answer
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.)
No comments:
Post a Comment