\documentclass[border=0.5cm]{standalone}
\usepackage{tikz, amsmath, bm}
\usetikzlibrary{positioning, colorbrewer}

\begin{document}
\begin{tikzpicture}

% Matrix A
% [[1, 1/2],
%  [1/2, 1]]
\draw[->, thick] (-1, 0) -- (1, 0);
\node at (0, 0) [above, shift = {(0, +0.2)}] {$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$};
\node at (0, 0) [below, shift = {(0, -0.2)}] {${{!}}A{{!}} = ad - bc$};

\begin{scope}[xshift = -4cm]
    % Circle
    \clip (0, 0) circle (2.5);
    \filldraw[fill = Set1-I!3, dotted] (0, 0) circle (2.5);

    % Unit Square
    \fill[Set1-C!25] (0, 0) -- (1, 0) -- (1, 1) -- (0, 1) -- cycle;
    \node at (1/2, 1/2) {$1$};

    % Original Vertical Lines
    \foreach \x in {-10, -9, ..., 10} {
        \draw[Set1-A] (\x, -10) -- (\x, 10);
    }
    % Original Horizontal Lines
    \foreach \y in {-10, -9, ..., 10} {
        \draw[Set1-B] (-10, \y) -- (10, \y);
    }

    % Origin
    \node at (0, 0) [circle, fill, inner sep = 1] {};

    % Vectors e1 and e2
    \draw[->, thick] (0, 0) -- (1, 0) node[near end, below, shift = {(0.4, -0.2)}, fill = Set1-I!3] {$\bm{e}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$};
    \draw[->, thick] (0, 0) -- (0, 1) node[near end, left,  shift = {(-0.2, 0.4)}, fill = Set1-I!3] {$\bm{e}_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$};
\end{scope}

% [x, y] maps to [x + y/2, x/2 + y]
\begin{scope}[xshift = +4cm]
    % Circle
    \clip (0, 0) circle (2.5);
    \filldraw[fill = Set1-I!3, dotted] (0, 0) circle (2.5);

    % Transformed Square
    \fill[Set1-C!25] (0, 0) -- (1, 1/2) -- (3/2, 3/2) -- (1/2, 1) -- cycle;
    \node at (13/16, 13/16) {${{!}}A{{!}}$}; % slightly away from the middle (3/4, 3/4)

    % Transformed Vertical Lines
    \foreach \x in {-10, -9, ..., 10} {
        \draw[Set1-A] (\x - 5, \x/2 - 10) -- (\x + 5, \x/2 + 10);
    }
    % Transformed Horizontal Lines
    \foreach \y in {-10, -9, ..., 10} {
        \draw[Set1-B] (-10 + \y/2, -5 + \y ) -- (10 + \y/2, 5 + \y);
    }

    % Origin
    \node at (0, 0) [circle, fill, inner sep = 1] {};

    % Vectors Ae1 and Ae2
    \draw[->, thick] (0, 0) -- (1, 1/2) node[midway, below right, fill = Set1-I!3] {$A\bm{e}_1 = \begin{bmatrix} a \\ c \end{bmatrix}$};
    \draw[->, thick] (0, 0) -- (1/2, 1) node[midway, above left,  fill = Set1-I!3] {$A\bm{e}_2 = \begin{bmatrix} b \\ d \end{bmatrix}$};
\end{scope}

\end{tikzpicture}
\end{document}