Transformations using matrices

Welcome to the World of Matrix Transformations!

Have you ever wondered how computer graphics in your favorite video games move characters around the screen? Or how photo editors "stretch" or "rotate" your pictures? Underneath all those buttons and sliders is the power of Matrix Algebra! In this chapter, we are going to learn how to use 2x2 matrices as "mathematical machines" that move, flip, and resize shapes on a coordinate grid. Don't worry if this seems a bit abstract at first—we'll break it down step-by-step!

1. The Basics: Points as Column Vectors

Before we can transform anything, we need to speak the right "language." In matrix transformations, we represent a point \( (x, y) \) as a column vector:

\( \begin{pmatrix} x \\ y \end{pmatrix} \)

When we multiply a 2x2 matrix by this column vector, we get a new column vector, which represents the image (the new position) of that point.

The Golden Rule: Matrix \( \times \) Object \( = \) Image
\( \mathbf{M} \mathbf{v} = \mathbf{v'} \)

Quick Review: How to Multiply

To find the new \( x \), multiply the top row. To find the new \( y \), multiply the bottom row:
\( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} \)

Takeaway: A matrix acts like a "rule" that tells every point on a shape exactly where to move.

2. The "Unit Vector" Trick (How to find any matrix)

Struggling to remember which matrix does what? There is a simple secret! Every transformation matrix is made up of what happens to the two "building blocks" of the grid:

1. The vector \( \mathbf{i} \): \( \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) (one step right)
2. The vector \( \mathbf{j} \): \( \begin{pmatrix} 0 \\ 1 \end{pmatrix} \) (one step up)

If you know where these two points land after a transformation, you have the matrix! The first column is where \( \mathbf{i} \) goes, and the second column is where \( \mathbf{j} \) goes.

Did you know? This is called "Linear Transformation" because the origin \( (0,0) \) never moves, and straight lines always stay straight!

3. Common Geometrical Transformations

Reflections

Imagine placing a mirror on the graph. The matrix "flips" the shape over the mirror line.

• Reflection in the x-axis: \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) (y-coordinates swap signs)
• Reflection in the y-axis: \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \) (x-coordinates swap signs)
• Reflection in \( y = x \): \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) (x and y swap places)
• Reflection in \( y = -x \): \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \)

Rotations

These matrices turn the shape around the origin \( (0,0) \). In math, positive angles move anti-clockwise!

The general matrix for a rotation through angle \( \theta \) is:
\( \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \)

Example: For a 90° anti-clockwise rotation, \( \cos(90)=0 \) and \( \sin(90)=1 \), giving us: \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \).

Enlargements and Stretches

These change the size of the shape.

• Enlargement (Centre (0,0), Scale Factor \( k \)): \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \)
• Stretch parallel to x-axis (Scale Factor \( k \)): \( \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \)
• Stretch parallel to y-axis (Scale Factor \( k \)): \( \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \)

Takeaway: If a matrix has 0s on the diagonal (the "off-diagonal"), it usually involves a reflection or rotation. If it has 0s in the corners, it's usually a stretch or enlargement.

4. Combining Transformations (The "Socks and Shoes" Rule)

What if we want to reflect a shape AND then rotate it? We use matrix multiplication!

If transformation \( \mathbf{A} \) is followed by transformation \( \mathbf{B} \), the combined matrix is \( \mathbf{BA} \).

Wait, why is it backward?
Think of it like putting on socks then shoes. If the vector is \( \mathbf{v} \):
1. First, apply A: \( \mathbf{A} \mathbf{v} \)
2. Then, apply B to the result: \( \mathbf{B}(\mathbf{A} \mathbf{v}) \)
This is why we write \( \mathbf{BA} \). The transformation closest to the vector happens first!

Common Mistake: Students often multiply in the order they read (A then B). Always write them right-to-left!

5. The Determinant and Area

One of the coolest things about matrices is that they tell you how much the area of a shape changes.

The area scale factor of a transformation is the absolute value of the determinant of the matrix.

\( \text{New Area} = |\text{det}(\mathbf{M})| \times \text{Old Area} \)

Quick Reminder: For matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), the determinant is \( ad - bc \).

Note: If the determinant is negative, it just means the shape has been "flipped" (reflected) as well as resized.

Takeaway: If the determinant is 1, the area doesn't change (like in a simple rotation or reflection).

6. The Inverse: Going Backwards

If a matrix \( \mathbf{M} \) transforms an object to an image, the inverse matrix \( \mathbf{M}^{-1} \) transforms the image back to the original object.

Don't worry if this seems tricky: You only need to find the inverse of a 2x2 matrix using the standard formula:
\( \mathbf{M}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \)

If the determinant is 0, the matrix is singular. This means the transformation squashes the whole shape into a single line or point—you can't "undo" that, so there is no inverse!

Key Takeaway Summary:
1. Use column vectors for points.
2. Columns of a matrix are the images of (1,0) and (0,1).
3. Multiplication order matters: \( \mathbf{BA} \) means A then B.
4. Determinant is the Area Scale Factor.
5. Inverse "undoes" the transformation.