Further Mathematics | Transformations using matrices

Welcome to Transformations using Matrices!

Hello! This chapter is where the geometry you learned previously meets the power of matrices. Don't worry if the term "transformation" sounds complicated—it just means moving or changing a shape. Using 2x2 matrices allows us to perform rotations, reflections, and much more, all through simple multiplication.

Why is this important? In areas like computer graphics, engineering, and physics, matrices are the fundamental tools used to calculate how objects move, scale, or distort in space. Mastering this chapter gives you a powerful algebraic tool for solving geometric problems!

1. The Basics: Mapping Points

1.1 What is a Transformation Matrix?

A transformation matrix, \( \mathbf{M} \), is a 2x2 matrix that, when multiplied by a position vector (a point), transforms it into a new position vector (the image point).

We represent a point \(P\) with coordinates \((x, y)\) as a column vector \( \mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix} \).

The transformation process is always:

$$ \mathbf{M} \mathbf{p} = \mathbf{p}' $$

Where \(\mathbf{p}'\) is the image point, \((x', y')\).

In simple terms: Think of the matrix \(\mathbf{M}\) as a little machine. You put the coordinates of your original point in, and the machine spits out the coordinates of the new, transformed point!

1.2 Step-by-Step Example of Mapping

Let the transformation matrix be \( \mathbf{T} = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \). Find the image of the point \( A(4, -1) \).

1. Write the point A as a column vector: \( \mathbf{a} = \begin{pmatrix} 4 \\ -1 \end{pmatrix} \).
2. Perform the matrix multiplication: $$ \mathbf{a}' = \mathbf{T} \mathbf{a} = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 4 \\ -1 \end{pmatrix} $$
3. Calculate the resulting vector: $$ \mathbf{a}' = \begin{pmatrix} (2)(4) + (1)(-1) \\ (0)(4) + (3)(-1) \end{pmatrix} = \begin{pmatrix} 8 - 1 \\ 0 - 3 \end{pmatrix} = \begin{pmatrix} 7 \\ -3 \end{pmatrix} $$

The image of \(A(4, -1)\) is \(A'(7, -3)\).

2. Finding the Matrix of a Transformation

Don't worry about memorizing every single transformation matrix straight away. There is a genius trick that allows you to derive any 2D transformation matrix, \(\mathbf{M}\).

2.1 The Basis Vectors Trick

All 2D transformations (with the origin remaining fixed) are defined by what happens to the two fundamental basis vectors:

• The vector along the x-axis: \( \mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \)
• The vector along the y-axis: \( \mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \)

The transformation matrix \(\mathbf{M}\) is constructed by taking the image of \(\mathbf{i}\) as the first column and the image of \(\mathbf{j}\) as the second column.

If transformation T maps \(\mathbf{i} \to \begin{pmatrix} a \\ c \end{pmatrix}\) and \(\mathbf{j} \to \begin{pmatrix} b \\ d \end{pmatrix}\), then:

$$ \mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$

Did you know? This is why we only need 2x2 matrices for 2D transformations centered at the origin. If the origin moves (translation), we need to use a different system (called homogeneous coordinates, which are usually covered in advanced topics, so we stick to origin-based transformations here!).

2.2 Example: Finding the Matrix for Reflection in \(y = -x\)

1. Where does \(\mathbf{i} = (1, 0)\) go?
Reflecting \((1, 0)\) in the line \(y = -x\) swaps the coordinates and changes the signs.
Image of \(\mathbf{i}\) is \((0, -1)\). This is the first column: \(\begin{pmatrix} 0 \\ -1 \end{pmatrix}\).

2. Where does \(\mathbf{j} = (0, 1)\) go?
Reflecting \((0, 1)\) in the line \(y = -x\).
Image of \(\mathbf{j}\) is \((-1, 0)\). This is the second column: \(\begin{pmatrix} -1 \\ 0 \end{pmatrix}\).

3. Assemble the matrix: $$ \mathbf{M} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} $$

3. Key Standard Transformations

You must be able to recognize and reproduce the matrices for these standard 2D transformations, all centered at the origin \((0, 0)\).

3.1 Rotations (R)

Rotations are anti-clockwise unless specified otherwise. If rotating clockwise by angle \(\theta\), use \(-\theta\) in the formula.

Rotation by angle \(\theta\) anti-clockwise about the origin: $$ \mathbf{R} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $$

Example: Rotation of \(90^\circ\) anti-clockwise (\(\theta = 90^\circ\)): $$ \mathbf{R}_{90} = \begin{pmatrix} \cos 90^\circ & -\sin 90^\circ \\ \sin 90^\circ & \cos 90^\circ \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$

3.2 Reflections (R)

Reflections map points onto their mirror images across a line (the mirror line).

• Reflection in the x-axis (\(y=0\)): $$ \mathbf{R}_{x} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
• Reflection in the y-axis (\(x=0\)): $$ \mathbf{R}_{y} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $$
• Reflection in the line \(y=x\): (Swaps x and y) $$ \mathbf{R}_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$
• Reflection in the line \(y=-x\): (Swaps x and y, and negates both) $$ \mathbf{R}_{y=-x} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} $$

3.3 Enlargements (E)

An enlargement is a scaling, centered at the origin, by a scale factor \(k\). Every coordinate is multiplied by \(k\).

Enlargement, scale factor \(k\), centre origin: $$ \mathbf{E} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $$

Note: If \(k\) is negative, it's an enlargement combined with a \(180^\circ\) rotation. If \(k=1\), the matrix is the Identity Matrix \( \mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \). The Identity Matrix does nothing to the shape!

3.4 Stretches (S)

A stretch changes the dimensions parallel to one axis only.

• Stretch parallel to the x-axis (y-coordinate is fixed) with factor \(k\): $$ \mathbf{S}_{x} = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} $$
• Stretch parallel to the y-axis (x-coordinate is fixed) with factor \(k\): $$ \mathbf{S}_{y} = \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} $$

3.5 Shears (Sh)

A shear slides a shape parallel to one axis, where the distance slid depends on the distance from the other axis.

• Shear parallel to the x-axis: The x-coordinate changes by \(k\) times the y-coordinate. (Lines parallel to the x-axis are invariant.) $$ \mathbf{Sh}_{x} = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} $$
• Shear parallel to the y-axis: The y-coordinate changes by \(k\) times the x-coordinate. (Lines parallel to the y-axis are invariant.) $$ \mathbf{Sh}_{y} = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} $$

4. Composite Transformations

In the real world, objects often undergo multiple movements. A composite transformation is simply one transformation followed by another.

4.1 Multiplication Order

If a shape is transformed by transformation \(T_1\) followed by transformation \(T_2\), and their respective matrices are \(\mathbf{M}_1\) and \(\mathbf{M}_2\), the combined transformation matrix, \(\mathbf{M}\), is found by multiplying the matrices.

$$ \mathbf{M} = \mathbf{M}_2 \mathbf{M}_1 $$

Crucial Point (Common Mistake Alert!): Matrix multiplication is performed in reverse order of the transformation applied!

If you apply \(T_1\) first, its matrix \(\mathbf{M}_1\) must be written closest to the position vector \(\mathbf{p}\): $$ \mathbf{p}' = \mathbf{M}_2 (\mathbf{M}_1 \mathbf{p}) $$

4.2 Example: Combining Transformations

Find the matrix for a reflection in the line \(y=x\), followed by a rotation of \(90^\circ\) clockwise about the origin.

1. Define the matrices (Clockwise \(90^\circ\) is \(\theta = -90^\circ\)):
Reflection \(T_1\): \( \mathbf{M}_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
Rotation \(T_2\): \( \mathbf{M}_2 = \begin{pmatrix} \cos(-90) & -\sin(-90) \\ \sin(-90) & \cos(-90) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \)

2. Calculate the composite matrix \(\mathbf{M} = \mathbf{M}_2 \mathbf{M}_1\): $$ \mathbf{M} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} (0)(0) + (1)(1) & (0)(1) + (1)(0) \\ (-1)(0) + (0)(1) & (-1)(1) + (0)(0) \end{pmatrix} $$

$$ \mathbf{M} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$

Observation: This composite transformation results in a reflection in the x-axis!

5. Area Scale Factor and Inverse Transformations

5.1 The Determinant and Area

The determinant of a transformation matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), denoted \(\det(\mathbf{M})\) or \(|\mathbf{M}|\), tells us the area scale factor of the transformation.

$$ \det(\mathbf{M}) = ad - bc $$

If a shape has original area \(A\), the new area \(A'\) is: $$ A' = |\det(\mathbf{M})| \times A $$ We use the absolute value because area is always positive.

• If \(|\det(\mathbf{M})| > 1\), the shape is enlarged.
• If \(|\det(\mathbf{M})| < 1\), the shape is shrunk.
• If \(|\det(\mathbf{M})| = 1\), the area is unchanged (as in rotations, reflections, and shears).

Singular Matrices: If \(\det(\mathbf{M}) = 0\), the matrix is singular. This means the transformation maps the entire plane onto a line or a single point. The area becomes zero, and the transformation is irreversible.

5.2 Inverse Transformations

If a transformation \(\mathbf{M}\) maps \(P \to P'\), the inverse transformation, \(\mathbf{M}^{-1}\), maps \(P' \to P\).

An inverse matrix exists only if the determinant is non-zero (i.e., the matrix is non-singular).

Recall the formula for the inverse 2x2 matrix: $$ \mathbf{M}^{-1} = \frac{1}{\det(\mathbf{M})} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$

The inverse transformation "undoes" the original transformation. For example, the inverse of a rotation by \(+30^\circ\) is a rotation by \(-30^\circ\).

6. Invariant Points and Lines

When a transformation is applied, some points might not move, and some lines might remain in the same position (even if the points on them move). These are key features of transformations.

6.1 Finding Invariant Points

An invariant point \(\mathbf{p}\) is a point such that the transformation maps it onto itself: \(\mathbf{M} \mathbf{p} = \mathbf{p}\).

Let \( \mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) and \( \mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix} \).

We solve the system of equations: $$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} $$

This can be rewritten as: $$ \begin{pmatrix} a-1 & b \\ c & d-1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$

Example: For any rotation or enlargement centered at the origin, the origin \((0, 0)\) is always the only invariant point.

6.2 Finding Invariant Lines

An invariant line is a line where every point on the line is mapped back onto the same line (though the points themselves might have moved along the line).

If \(y = mx + c\) is an invariant line, then for any point \((x, y)\) on the line, the image \((x', y')\) must also satisfy \(y' = mx' + c\).

This often requires setting up simultaneous equations and identifying cases where the result holds true for a range of x-values. This is often the most algebraically demanding part of the topic.