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.

Solution : A solution to an equation is a choice of variables that when substituted into the equation results in a true statement.

Square root : The square root of $x$ denoted by $\sqrt{x}$ is the positive number such that $(\sqrt{x})^2=\sqrt{x}\cdot \sqrt{x}=x.$ For example $\sqrt{9}=3.$

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

Linear combination : A linear combination of vectors $x_1, x_2,\ldots, x_n$ in a vector space is any vector of the form $\lambda_1x_1+\lambda_2x_2+\ldots+\lambda_nx_n$ where $\lambda_1,\lambda_2,\ldots,\lambda_n$ are real numbers.

Linear independence : Vectors $x_1, x_2,\ldots, x_n$ in a vector space are called linearly independent if $\lambda_1 x_1+\lambda_2x_2+\ldots+\lambda_nx_n=0$ implies $\lambda_1=\lambda_2=\ldots=\lambda_n=0.$ In other words vectors are linearly independent if the only linear combination of the zero vector is the trivial one in which all coefficients are zero.

Vector : Most commonly vector refers to a matrix with one column $\begin{pmatrix}x_1\\ \vdots \\x_n\\\end{pmatrix}.$ In general a vector is an element of a vector space.

Binomial coefficient : A binomial coefficient $\binom{n}{k}$ is defined as $\frac{n!}{k!(n-k)!}$ and equals the number of different sets of $k$ objects one can pick out of $n$ objects.

Factorial : The factorial of a positive integer $n$ written as $n!$ is defined by $n!=1\cdot2\cdot3\ldots\cdot n.$ For example we have $5!=1\cdot2\cdot3\cdot4\cdot 5=120.$ $n!$ is the number of different permutations of $n$ objects.