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I'm given the vectors $v_1=(4,1)$, $v_2=(-7,-8)$, and I'm trying to figure out see if they form a basis for $\mathbb{R}^2$.

I think that it is a basis because $v_1$ and $v_2$ are independent of each other but I'm not sure if it's that easy. Am I on the right track?

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    @Cascadia: If you are checking to see if it is linearly independent, you would set $\alpha(4,1)+\beta(-7.-8)=(0,0)$ and see if there are any solutions other than $\alpha=\beta=0$ (if there are, it is dependent, if there aren't, it is independent). But that part of my comment was about seeing they span. Here you want to see if for *any* $a$ and $b$, you can always solve $\alpha(4,1) + \beta(-7,-8) = (a,b)$.2011-03-19

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I think you mean that you are given a pair of vectors $\{(4,1),(-7,-8)\}$ and asking whether or not they form a basis for $\mathbb{R}^2$. If you are allowed to use the fact that the dimension of a vector space is well-defined, all you need to prove is that the vectors are linearly independent or that they span the space (as either of these, the fact that dim$(\mathbb{R}) = 2$), implies the other); otherwise you must prove both.

To prove that the vectors are linearly independent, try to solve the equation $a(4,1)+b(-7,-8)=(0,0)$ and show that no solution exists.

To prove that the vectors span the space, show that $(1,0)$ and $(0,1)$ can be written as a linear combination of the vectors you are given, thus any vector in $\mathbb{R}^2$ can be written as such.

Hope that helps.