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I am given these two vectors

(1,2), (2,1)

and i know that for a set of vectors to form a basis, they must be linearly independent and they must span all of R^n

I know that these two vectors are linearly independent, but i need some help determining whether or not these vectors span all of R^2

So far i have the equation below

a(1,2) + b(2,1) = (x,y)

i am assuming that i must solve for x and y by setting up a system of equations.. but i have a little trouble in doing that with this particular equation,

i used row reduction and got y = -3b and x = a + 2b

now that i got x and y, does that mean that the vectors form a basis? or is there something else that i am missing here?

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2 Answers

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First of all, note that if you know that the two vectors are linearly independent, and live in a two dimensional space they must span (otherwise the space really wasn't two dimensional).

To see this explicitly, take some $(x,y)$ in $\mathbb{R}^2$ and solve the system $$ a(1,2)+b(2,1)=(x,y) $$ for $a$ and $b$, not $x,y$ which are arbitrary here (I think you are trying to do this). This is a system of two equations in two unknowns with solution $$ a=\frac{1}{3}(2y-x)\\ b=\frac{1}{3}(2x-y) $$

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The vectors $(1,2)$ and $(2,1)$ are linearly independent, and because $\text{dim}(\mathbb{R}^2)=2$ we can conclude that $\text{span}((1,2),(2,1))=\mathbb{R}^2$, i.e., the set $\{(1,2),(2,1)\}$ is a basis for $\mathbb{R}^2$

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