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 $n=1$.)

Equation : An equation is a mathematical statement in which two expressions are written with an equal sign in between. A solution of an equation is a set of variables that makes the statement a true statement.

Induction : Induction is a common techniques for proofing mathematical results that are dependent on a natural number $n.$ It consists of first proving the result for a specific $n$, often $n=0$ or $n=1$ (the induction base) and then proving that if it the results holds for all numbers smaller than $n$ that it will then also hold true for $n$ (the induction step.)

Proof : A proof is a logical deduction of a result starting only with the assumptions of the result.

Squared : $x$ squared refers to the number $x^2=x\cdot x.$ For example 3 squared equals 9.