Space generated by vectors

Question

I have a doubt: can you say, for sure, that every space generated by two linear independente vectors with two components generate $\mathbb{R^2}$?

For example: $L$ {$(1,1),(0,2)$} = $\mathbb{R^2}$

(which means that this type of vectors are always a basis of $\mathbb{R^2}$)

Am I correct to admit this as always true?

Thanks!

athul777 · Accepted Answer · 2016-05-24 10:16:56Z

2

Yes, since the dimension of $\mathbb{R}^{2}$ is $2$. So any two linearly independent vectors will always be a basis for $\mathbb{R}^{2}$.

answered May 24, 2016 at 10:16

athul777

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$\begingroup$ But the vectors need to have only 2 components $(x,y)$ right? $\endgroup$
– Granger Obliviate
May 24, 2016 at 10:19
$\begingroup$ Yes. But can you clarify what you mean by "need to have only two components"? What other configurations are you thinking of? $\endgroup$
– athul777
May 24, 2016 at 10:24
$\begingroup$ Yes, I'm thinking of having for example the space $L$ {$(1,1,2),(0,2,1)$}. This is indeed a space of dimension 2 but it isn't $\mathbb{R^2}$, right? $\endgroup$
– Granger Obliviate
May 24, 2016 at 10:31
1

$\begingroup$ That isn't $\mathbb{R}^{2}$, but the space $L$ is isomorphic to $\mathbb{R}^{2}$. $\endgroup$
– athul777
May 24, 2016 at 10:36
$\begingroup$ Exactly, same dimension! Thank you very much! $\endgroup$
– Granger Obliviate
May 24, 2016 at 10:54

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