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?
Best Answer
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 $$ a=\frac{1}{3}(2y-x)\\ b=\frac{1}{3}(2x-y) $$