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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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Yes, you are. If you already know that every basis for $\mathbb{R}^2$ has two elements, and that every linearly independent set with $2$ elements is a basis for $\mathbb{R}^2$, then you can show that $v_1$ and $v_2$ form a basis (there is more than one basis, so it is incorrect to call it "the" basis) by simply showing they are linearly independent. If you don't know that yet, then you need to show both that they are linearly independent, and that they span $\mathbb{R}^2$ (every $(a,b)$ can be written as $\alpha(4,1) + \beta(-7,-8)$ for suitably chosen scalars $\alpha,\beta$). – Arturo Magidin Mar 19 '11 at 3:39
Couldn't I also use the determinant of the matrix to see if it's independent or not? – Cascadia Mar 19 '11 at 4:06
@Cascadia: I didn't say anything about how to show they are linearly independent. I just said to show they are linearly independent. Or did you perhaps mean to comment on Alex's answer? – Arturo Magidin Mar 19 '11 at 4:10
I miss read your comment when you said I could find out if it's indie or dependent by α(4,1)+β(−7,−8). I thought I would actually have to set it equal to a 0 vector. – Cascadia Mar 19 '11 at 4:20
Yes, you could use the determinant technique, I just gave my answer because I though it was more intuitive if you're not familiar with the properties of matrices. – Alex Becker Mar 19 '11 at 4:26
up vote 0 down vote accepted

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.

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