 Solution

For two nonparallel vectors $v_1,v_2\in {\mathbb{R}}^2$, a nontrivial relation $a{v}_1+b{v}_2=0$ 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.