Basis : The basis of a vector space is a set of linearly independent vectors, so that every vector of the vector space can be written as a linear combination of the vectors in the basis. For example $\{\left(\begin{smallmatrix}1\\0\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\1\\0\\\end{smallmatrix}\right),\left(\begin{smallmatrix}0\\0\\1\\\end{smallmatrix}\right)\}$ is a basis of $\mathbb{R}^3.$ In a prove by induction basis refers to the first step in which you have to prove that the result is true for a particular number $n$ (usually $n=0$ or $n=1$.)