Matrix and Transformation Connection

What is the Connection between Matrices and Geometric Transformations?

A $2 \times 2$ matrix can be associated with transformation operations on any point in the Cartesian plane.

A point in the Cartesian plane, often symbolized by the ordered pair $(x,y)$, can also be symbolized by the position vector $\begin{pmatrix} x \\ y \end{pmatrix}$. This position vector notation will be frequently used in discussing the connection between matrices and transformations.

If a point $P(x,y)$ is transformed by the matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, its image $P'(x',y')$ is obtained from matrix multiplication:

Thus, $x' = ax + by$ and $y' = cx + dy$.

Multiplying a Matrix by a Position Vector

If $\begin{pmatrix} x \\ y \end{pmatrix}$ represents any point in the Cartesian plane, find the product of $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$.

Alternative Solution:

The matrix product is:

It can be observed that the point $(x,y)$ is transformed by the matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ into the point $(-y,x)$. This is the formula for a $90^\circ$ counter-clockwise rotation about the origin.

Multiplying a Matrix by Three Points Simultaneously

Find the image of $\triangle ABC$, with vertices $A(1,1)$, $B(4,1)$, and $C(4,2)$ transformed by the matrix $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$.

Alternative Solution:

First, we can write the coordinates of the points as columns of a matrix, i.e., $\begin{pmatrix} 1 & 4 & 4 \\ 1 & 1 & 2 \end{pmatrix}$ (Columns A, B, C).

Next, multiply this matrix from the left by $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$.

The result of the transformation is a new triangle $\triangle A'B'C'$ with vertices $A'(-1,-1)$, $B'(-4,-1)$, and $C'(-4,-2)$.

The matrix $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ represents a $180^\circ$ rotation about the origin.

Exercises

1. Find the product of $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$. What transformation does this matrix represent?
2. A transformation is associated with the matrix $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$. Find the image of a triangle with vertices $A(2,0)$, $B(2,1)$, and $C(0,1)$ under this transformation!

Key Answers

1. $$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} (0)(x) + (1)(y) \\ (-1)(x) + (0)(y) \end{pmatrix} = \begin{pmatrix} y \\ -x \end{pmatrix}$$

   The point $(x,y)$ is transformed into $(y,-x)$.

   This is a $-90^\circ$ (or $270^\circ$) clockwise rotation about the origin.

2. Transformation matrix $M = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$.

   Vertices: $A(2,0)$, $B(2,1)$, $C(0,1)$.

   Point matrix: $\begin{pmatrix} 2 & 2 & 0 \\ 0 & 1 & 1 \end{pmatrix}$.

   $$\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 2 & 0 \\ 0 & 1 & 1 \end{pmatrix} = \begin{pmatrix} (1)(2)+(2)(0) & (1)(2)+(2)(1) & (1)(0)+(2)(1) \\ (0)(2)+(1)(0) & (0)(2)+(1)(1) & (0)(0)+(1)(1) \end{pmatrix}$$

   $$= \begin{pmatrix} 2+0 & 2+2 & 0+2 \\ 0+0 & 0+1 & 0+1 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 2 \\ 0 & 1 & 1 \end{pmatrix}$$

   Image vertices: $A'(2,0)$, $B'(4,1)$, $C'(2,1)$.

   (This transformation is known as a shear)