= Solution
For two nonparallel vectors $v_1,v_2\in\mathbb R^2$, a nontrivial relation $av_1+bv_2=0$ would make one a scalar multiple of the other. They are therefore <linearly independent>. Two independent vectors in a two-dimensional <vector space> form a <basis>.
Likewise, three vectors in $\mathbb R^3$ that are not contained in one plane through the origin cannot obey a nontrivial <linear dependence>: such a relation would express one as a combination of the other two, or already make that pair dependent. Thus they are independent and form a <basis> of $\mathbb R^3$. The plane condition here is linear coplanarity with the origin, as required for vectors, rather than affine coplanarity of three arbitrary points.
Back to article page
