Introduction
Points on a number plane that can be represented by a coordinate (x,y) can also be represented by the column matrix \begin{bmatrix} x \\ y \end{bmatrix}.
Multiple points can be represented in the one matrix. The coordinates (3,6) , (-2, 4) and (4,5) can be represented by the following matrix: \begin{bmatrix} &3 &-2 &4 \\ &6 &4 &5 \end{bmatrix}
Ideas
Transform points
Representing points by column matrices is useful when we want to move all the points in the plane in certain ways. The matrix operations of addition and multiplication give us two kinds of these transformations in the plane. We follow specific algorithms to perform these operations on matrices.
Starting with the point ( x, y) we can slide the point a units in the x-direction and b units in the y-direction by adding the column matrix as shown below.\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} x + a \\ y + b \end{bmatrix}
This kind of transformation is called a translation.
We can multiply a column matrix, representing a point, by a 2 \times 2 transformation matrix as follows. \begin{bmatrix} a & c \\ b &d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax +cy \\ bx + dy \end{bmatrix}
The algorithm for multiplying a matrix representing 3 points by a 3 \times 2 transformation matrix looks like this: \begin{bmatrix} a & c \\ b &d \end{bmatrix} \times \begin{bmatrix} A &B &C \\ X &Y &Z \end{bmatrix} = \begin{bmatrix} a A + b X &a B+b Y &a C + b z \\ c A + d X &c B + d Y &c C +d Z \end{bmatrix}
Depending on how the transformation matrix is chosen, we can use it to reflect points across any line, dilate vertically and horizontally by specified dilation factors and rotate about the origin.
Reflections of the plane in the x-axis and y-axis respectively are given by the transformation matrices: R_x = \begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix}, \qquad R_y = \begin{bmatrix} -1 &0 \\ 0 &1 \end{bmatrix} which we multiply the matrix of points with to find the coordinates of the reflected points.
The following matrix multiplication will dilate (stretch or squash) the point (x,y) by a factor of a with respect to the x-axis and by a factor of b with respect to the y-axis: \begin{bmatrix} a &0 \\ 0 &b \end{bmatrix} \times \begin{bmatrix} x \\ y \end{bmatrix}
If points in the plane are to be rotated anticlockwise through an angle \theta, the transformation matrix is: T_\theta = \begin{bmatrix} \cos \theta &-\sin \theta \\ \sin \theta &\cos \theta \end{bmatrix} which we multiply the matrix of points with to find the coordinates of the rotated points.
Examples
Example 1
A set of three points are given in the matrix below. \begin{bmatrix} -3 &-1 &-7 \\ -4 &-8 &-2 \end{bmatrix}
They are all translated using the transformation matrix: \begin{bmatrix} 0 &0 &0 \\ 7 &7 &7 \end{bmatrix}
Calculate the coordinates of the newly translated points: \begin{bmatrix} -3 &-1 &-7 \\ -4 &-8 &-2 \end{bmatrix} + \begin{bmatrix} 0 &0 &0 \\ 7 &7 &7 \end{bmatrix} = \begin{bmatrix} ⬚ &⬚ &⬚ \\ ⬚ &⬚ &⬚ \end{bmatrix}
Example 2
Consider the points: A(-2, 1), B(-7, 1), C(-7, 3), D(-2, 3).
Plot the four points on the graph below:
State the coordinates of the four points if we translated each 10 units right.
We can use a simple transformation matrix to find the location of the new points. Complete the values in the transformation matrix below. \begin{bmatrix} -2 &-7 &-7 &-2 \\ 1 &1 &3 &3 \end{bmatrix} + \begin{bmatrix} ⬚ &⬚ &⬚ &⬚ \\ ⬚ &⬚ &⬚ &⬚ \end{bmatrix} = \begin{bmatrix} 8 &3 &3 &8 \\ 1 &1 &3 &3 \end{bmatrix}
Example 3
Consider the following:
By considering the two points with array: \begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix} determine the transformation matrix that reflects points over the x-axis.
Consider the points A, B, C in the graph below.
Use the transformation matrix from part (a) to reflect the points A, B, C about the x-axis.\begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix} \times \begin{bmatrix} -4 &-9 &-2 \\ 9 &0 &9 \end{bmatrix} = \begin{bmatrix} ⬚ &⬚ &⬚ \\ ⬚ &⬚ &⬚ \end{bmatrix}
By considering the two points with array: \begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix} determine the transformation that will rotate points 90\degree anticlockwise.
In part (b), we found the points A', B' C'. They are plotted in the graph below.
Use the transformation matrix from part (c) to rotate the points A', B', C' drawn above. \begin{bmatrix} 0 &-1 \\ 1 &0 \end{bmatrix} \times \begin{bmatrix} -4 &-9 &-2 \\ -9 &0 &-9 \end{bmatrix} = \begin{bmatrix} ⬚ &⬚ &⬚ \\ ⬚ &⬚ &⬚ \end{bmatrix}
Algorithm for multiplying 2 \times 2 matrices: \begin{bmatrix} a & c \\ b &d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} ax +cy \\ bx + dy \end{bmatrix}
Algorithm for multiplying 3 \times 2 matrices: \begin{bmatrix} a & c \\ b &d \end{bmatrix} \times \begin{bmatrix} A &B &C \\ X &Y &Z \end{bmatrix} = \begin{bmatrix} a A + b X &a B+b Y &a C + b z \\ c A + d X &c B + d Y &c C +d Z \end{bmatrix}
We can translate ( x, y) by a units in the x-direction and b units in the y-direction by adding the matrices:\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} x + a \\ y + b \end{bmatrix}
Reflection matrices for reflections in the x-axis and y-axis: R_x = \begin{bmatrix} 1 &0 \\ 0 &-1 \end{bmatrix}, \qquad R_y = \begin{bmatrix} -1 &0 \\ 0 &1 \end{bmatrix}
Dilation matrix by a factor of a in the x-direction and by a factor of b in the y-direction: \begin{bmatrix} a &0 \\ 0 &b \end{bmatrix}
Rotation matrix for rotating points anticlockwise through an angle \theta:T_\theta = \begin{bmatrix} \cos \theta &-\sin \theta \\ \sin \theta &\cos \theta \end{bmatrix}