From what I know, a basis is a linearly independent spanning set. And a spanning set is just all the linear combinations of the vectors.
Lets say we have the two vectors
$a =(1,2)$
$b =(2,1)$
So I will assume that the first step involves proving that the vectors are linearly independent
x(1,2)+ y(2,1) = (0,0)
After doing row reduction, i found out that the vectors are linearly independent because the only solutions for x and y are 0
I'm not sure what to do next. I know how to determine if a set of vectors forms a basis over a vectors space $R^n$, but to actually find a basis of a set of vectors is a little different for me.
any help will be appreciated.
Best Answer
You're done.
You're working in $\mathbb{R}^2$, so any basis of $\mathbb{R}^2$ will contain two linearly independent vectors. After showing that your system admits only the trivial solution, you're found that those two vectors are linearly independent (and since there's exactly two, form a spanning set). This is enough to find a basis.
Now, in general, if you have more than the required number of vectors to build a basis, you can figure out which ones are a linear combination of the others and remove them from the set in question. This is something many linear algebra texts have as exercises. You can even extend a linearly independent set of vectors to a basis for a vector space by means of Gram-Schmidt or some other process. Just have to be careful!