Select an object to transform (point, line, circle, function, unit square), then select a transformation.[br][br]You can explore the transformations by dragging the initial objects that are not constrained by a definition. These objects are displayed in green.[br][br]The transformed objects are displayed in red, and cannot be dragged, since they are dependent on the initial objects.[br][br]You can apply some [color=#2980b9][i]predefined transformations[/i][/color]: [color=#2980b9][i]reflections[/i][/color] about the axes, [color=#2980b9][i]shears[/i][/color], [color=#2980b9][i]dilations [/i][/color]and [color=#2980b9][i]rotations[/i][/color].[br][br]Or you can select a [color=#2980b9][i]Custom[/i][/color] transformation, and define yourself the matrix of the transformation [math]T=\begin{bmatrix} t_{11} & t_{12} \\ t_{21} & t_{22} \end{bmatrix} [/math] using the displayed sliders.
You can use matrices to compute the coordinates of transformed points from a given set, under a transformation represented by a matrix [math]T[/math].[br][br]If the coordinates of the given points are [math]\left(x_1,y_1\right),\left(x_2,y_2\right)[/math] and [math]\left(x_3,y_3\right)[/math], then the coordinate matrix is [math]M=\begin{bmatrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{bmatrix} [/math], and the coordinate matrix of the transformed points is obtained by [math]M'=T\cdot M[/math].
Consider the unit square in the app above. If you consider the coordinates of its vertices, listed anticlockwise starting from the origin, you can form the coordinate matrix of the vertices: [math]M=\begin{bmatrix} 0 & 1 & 1 & 0\\ 0 & 0 & 1& 1 \end{bmatrix} [/math].[br][br]Now select a [i]Custom[/i] transformation, and set the two elements of the main diagonal of the transformation matrix [math]T[/math] equal to 2, and the remaining elements equal to 0.[br]What transformation have you obtained?[br]Compare the areas of the unit square and the transformed square.[br][br]If you set the transformation matrix equal to [math]T=\begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix} [/math], what would be the area of the transformed figure?[br][br]Verify your conjecture by computing the coordinates of the vertices of the transformed figure, and determining its area.[br]
The list of transformations available in the app doesn't include [i]reflection about the origin[/i].[br]Can you obtain this transformation using one of the available options?[br]Explain your answer in detail.