I am reading Fulton's book on algebraic curves.In the second chapter they have defined what they call affine coordinate change map between two affine spaces.It is defined in the following manner:
Let $k$ be an algebraically closed field and $\mathbb A^n_k$ be the affine $n$-space over $k$.Then a polynomial map $T:\mathbb A^n\to \mathbb A^n$ is said to be an affine change of coordinates if $T=(T_1,T_2,...,T_n)$ where $T_i$ 's are degree $1$ polynomials of $k[X_1,X_2,...,X_n]$ and $T$ is bijective.
I want to understand what it intuitively means.Is it closely related to linear transformations?I want to get some idea about what it would look like for known spaces like $\mathbb R^2$ or $\mathbb R^3$.Can someone illustrate these things and explain to me?
We should work out what this is saying for $\Bbb{A}^3$ as you propose (n.b. that the field here doesn't really play any role - you could take it to be $\Bbb{R}$ if it is more familiar). In this case, an affine change of coordinates is given by $f:\Bbb{A}^3 \to \Bbb{A}^3$ where $f(x,y,z) = (f_1(x,y,z),f_2(x,y,z),f_3(x,y,z))$ are degree $1$ polynomials. So, the function is of the form $f_i(x,y,z) = a_ix+b_iy+c_i z +d_i$, and in vector notation this transformation is $$ f(x,y,z) = \begin{bmatrix} a_1&b_1&c_1\\ a_2&b_2&c_2\\ a_3&b_3&c_3 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} + \begin{bmatrix} d_1\\ d_2\\ d_3 \end{bmatrix}. $$ The composition is supposed to be bijective: this is the composition of $x\mapsto Ax$ followed by $y\mapsto y +d$, where $A$ is the $3\times 3$ matrix above and $d$ is the column vector $(d_1,d_2,d_3)^t$. $x\mapsto Ax$ is bijective if and only if the determinant is nonzero. $y\mapsto y+d$ is always bijective. So, these "affine transformations" are all of the form $$ f(x) = Ax+b $$ where here $A$ is a matrix and $x$ is a vector. Note that this does not use the field or the $3$-dimensional assumption. It is quite general. An affine transformation is therefore a composition of a linear (nonsingular) map and a translation.