Basis
Definition: The vectors v1, v2,..., vk in a vector space V are said to form a basis for V if (a) v1, v2,..., vk span V and (b) v1, v2,..., vk are linearly independent. Note* If v1, v2,..., vk form a basis for a vector space V, then they must be distinct and nonzero. Note** The above definition not only applies to a finite set of vectors, but also to an infinite set of vectors in a vector space.
Example 1
Let $ V = R^3 $. The vectors $ (\left(\begin{array}{cccc}a11&a12\\a21&a22\end{array}\right) $
