Pure Mathematics | Transformations using matrices

Unit FP1: Transformations Using Matrices

Introduction: Geometry Meets Algebra

Hello mathematicians! Welcome to the exciting world where geometry meets algebra. This chapter, Transformations using Matrices, is one of the most powerful and elegant topics in Further Pure Mathematics 1 (FP1).

Instead of describing movements and changes (like reflections, rotations, and stretching) using clunky equations, we will use simple 2x2 matrices. These matrices act like operational switches—input a coordinate, and the matrix spits out the new, transformed coordinate.

Don't worry if matrices felt a bit abstract before. Here, you'll see them in action, making complex geometric operations incredibly streamlined.

1. Representing Points and Applying Transformations

1.1. How Coordinates Become Vectors

In coordinate geometry, we use the pair \((x, y)\) to define a point. When using matrices, we represent this point as a position vector (a \(2 \times 1\) column matrix).

$$ \text{Point } (x, y) \text{ becomes the vector } \mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix} $$

1.2. The Transformation Equation

A transformation \(T\) is represented by a \(2 \times 2\) matrix, usually denoted by \(\mathbf{M}\). To find the new point, \((x', y')\), after the transformation, we simply multiply the matrix by the original vector:

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

In full:

$$ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix} $$

Key Takeaway: Matrix multiplication is the mechanism for transformation. We always write the matrix first, then the vector.

2. Standard Transformations and Their Matrices

FP1 focuses on transformations centered at or relating to the origin \((0, 0)\). These are sometimes called linear transformations. We need to memorize or be able to derive the matrices for the following common types.

2.1. Rotation About the Origin (0, 0)

A rotation takes a point and spins it around the origin by an angle \(\theta\).

• Rotation is positive for anti-clockwise (counter-clockwise) movement.
• Rotation is negative for clockwise movement.

The General Rotation Matrix (Anti-Clockwise by \(\theta\)):

$$ \mathbf{R} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $$

Example: Rotation of \(90^\circ\) anti-clockwise:

$$ \mathbf{R}_{90} = \begin{pmatrix} \cos 90^\circ & -\sin 90^\circ \\ \sin 90^\circ & \cos 90^\circ \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$

2.2. Enlargement (Scaling)

An enlargement increases or decreases the size of a shape. Since we are using 2x2 matrices, the centre of enlargement is always the origin \((0, 0)\).

If the scale factor is \(k\):

$$ \mathbf{E} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $$

Did you know? If \(k=1\), the matrix is the Identity Matrix \(\mathbf{I}\). This transformation does nothing! $$ \mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$

2.3. Reflection

A reflection flips a shape across a line (the mirror line).

Common Reflection Matrices:

1. Reflection in the x-axis (\(y=0\)): $$ \mathbf{M}_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
2. Reflection in the y-axis (\(x=0\)): $$ \mathbf{M}_y = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $$
3. Reflection in the line \(y=x\): (Swaps x and y coordinates) $$ \mathbf{M}_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$
4. Reflection in the line \(y=-x\): $$ \mathbf{M}_{y=-x} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} $$

2.4. Shear

A shear is a transformation that slides all points in one direction, parallel to a specific line, while keeping points on that line fixed (the invariant line).

If the shear factor is \(k\):

1. Shear parallel to the x-axis (x-axis is the invariant line):

   The \(y\) coordinate stays the same, the \(x\) coordinate shifts by \(k\) times the original \(y\) value. \((x, y) \to (x+ky, y)\)

   $$ \mathbf{S}_x = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} $$
2. Shear parallel to the y-axis (y-axis is the invariant line):

   The \(x\) coordinate stays the same, the \(y\) coordinate shifts by \(k\) times the original \(x\) value. \((x, y) \to (x, y+kx)\)

   $$ \mathbf{S}_y = \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} $$

3. Finding the Matrix of a Transformation

You won't always be given the matrix; sometimes you must derive it from a geometric description. This is arguably the most important skill in this chapter.

3.1. The Magic of Basis Vectors

Any \(2 \times 2\) matrix \(\mathbf{M}\) can be determined entirely by looking at what happens to two special points: the unit vectors along the axes. These are the basis vectors:

1. \(\mathbf{i}\): The point \((1, 0)\), represented as \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\).
2. \(\mathbf{j}\): The point \((0, 1)\), represented as \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\).

When you multiply a general matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) by \(\mathbf{i}\) and \(\mathbf{j}\):

$$ \mathbf{M} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} a \\ c \end{pmatrix} \quad \text{and} \quad \mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} b \\ d \end{pmatrix} $$

The Simple Trick:

The first column of the transformation matrix \(\mathbf{M}\) is the image of the vector \(\mathbf{i}\) (i.e., where \((1, 0)\) moves to).

The second column of the transformation matrix \(\mathbf{M}\) is the image of the vector \(\mathbf{j}\) (i.e., where \((0, 1)\) moves to).

$$ \mathbf{M} = \begin{pmatrix} \text{Image of } (1, 0) & \text{Image of } (0, 1) \\ \text{ (Column 1) } & \text{ (Column 2) } \end{pmatrix} $$

Example: A transformation reflects a shape in the line \(y = 2x\). If \((1, 0)\) maps to \((0.6, 0.8)\) and \((0, 1)\) maps to \((0.8, -0.6)\).

The matrix \(\mathbf{M}\) is simply: $$ \mathbf{M} = \begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix} $$

Key Takeaway: If you can figure out where \((1, 0)\) and \((0, 1)\) move, you know the entire transformation matrix.

4. Combining Transformations (Composition)

Often, a single operation is not enough. We might need to rotate a shape and then enlarge it. This is called a composition of transformations.

4.1. Order Matters!

If a transformation \(T_1\) is applied first, followed by a transformation \(T_2\), the combined transformation \(T\) is represented by the matrix product \(\mathbf{M}_2 \mathbf{M}_1\).

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

Analogy: Think of putting on socks and then shoes. You must do the socks (\(T_1\)) first, but the operation you write down is \(\text{Shoes} \times \text{Socks}\).

CRUCIAL RULE: Matrices are multiplied in the reverse order of application. The matrix closest to the vector \(\mathbf{x}\) is the one applied first.

$$ \mathbf{M}_2 (\mathbf{M}_1 \mathbf{x}) = (\mathbf{M}_2 \mathbf{M}_1) \mathbf{x} $$

Common Mistake Alert: Since matrix multiplication is generally not commutative (\(\mathbf{M}_1 \mathbf{M}_2 \neq \mathbf{M}_2 \mathbf{M}_1\)), rotating then reflecting usually gives a different result than reflecting then rotating. Pay close attention to the order specified in the question!

5. Area Scale Factor and the Determinant

5.1. The Determinant as an Area Factor

One of the most useful properties of a transformation matrix \(\mathbf{M}\) is its determinant, denoted \(\det(\mathbf{M})\) or \(|\mathbf{M}|\).

For a \(2 \times 2\) matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant is:

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

The absolute value of the determinant, \(|\det(\mathbf{M})|\), is the Area Scale Factor.

• If an original shape has area \(A_{original}\), the new shape will have area \(A_{new} = |\det(\mathbf{M})| \times A_{original}\).
• If \(\det(\mathbf{M})\) is positive, the orientation of the shape is preserved (e.g., rotation, enlargement).
• If \(\det(\mathbf{M})\) is negative, the orientation is reversed (e.g., reflection).

5.2. Singular Matrices

If the determinant is zero, \(\det(\mathbf{M}) = 0\), the matrix is called a singular matrix.

• If the Area Scale Factor is zero, it means the entire shape has collapsed onto a line or a single point.
• A singular transformation squashes a 2D area into a 1D line, meaning the inverse matrix does not exist (it is impossible to "un-squash" the shape back into 2D).

6. Inverse Transformations

The inverse transformation, represented by \(\mathbf{M}^{-1}\), is the transformation that "undoes" the effect of \(\mathbf{M}\). If \(\mathbf{M}\) rotates a shape \(90^\circ\) anti-clockwise, then \(\mathbf{M}^{-1}\) rotates it \(90^\circ\) clockwise.

6.1. Finding the Inverse Matrix

For a non-singular matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), where \(D = \det(\mathbf{M}) = ad - bc \neq 0\):

$$ \mathbf{M}^{-1} = \frac{1}{D} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$

Memory Aid for \(2 \times 2\) Inverses:

1. Find the Determinant \(D\).
2. Swap the entries on the main diagonal (\(a\) and \(d\)).
3. Change the sign of the entries on the off-diagonal (\(-b\) and \(-c\)).
3. Multiply the resulting matrix by \(1/D\).

Key Takeaway: If a transformation is followed by its inverse, the result is the Identity matrix (\(\mathbf{M} \mathbf{M}^{-1} = \mathbf{I}\)), meaning the shape returns to its original position.

Congratulations! Mastering these concepts—especially the relationship between transformations, basis vectors, determinants, and composition order—will give you a strong foundation for the rest of FP1. Keep practicing those matrix multiplications!