Number of parameters to specify an affine transformation in n dimensions

Question

In general, how many parameters does it take to specify an affine transformation in $n$ dimensions, and how does one go about proving this?

For example, in 2 dimensions it takes 6 parameters, and in 3 dimensions it takes 12 parameters.

Answer 1

An affine transformation is a linear transformation followed by a translation. The former is defined by an $n\times n$ matrix and the later is defined by a vector. So, you need $n+n^2=n(n+1)$ parameters to define an affine transformation on $\mathbb{R}^n$.

Answer 2

An affine transformation is a map of the form $x \mapsto Ax + b$. So read off from the $n^2$ independent entries of $A$ and $n$ independent entries of $b$ that the answer is $n^2 + n$. I assume here the underlying field is $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{Q}$, not something more exotic like a finite field.