Matrices for Transformations
CSEC Mathematics: The Geometry of Matrices
Essential Understanding: A transformation matrix is a mathematical "machine" that moves, stretches, rotates, or flips shapes on the coordinate plane. Master the four standard transformations and their matrices to solve geometric problems efficiently.
1. The Transformation Concept: \(M \times P = P'\)
Start by defining the transformation equation. In CSEC, we use a \(2 \times 2\) matrix (\(M\)) to move a point or shape (\(P\)) to its new image (\(P'\)).
The Object (\(P\))
Represented as a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\).
For a shape with multiple vertices, arrange them in a matrix with each column representing a point.
Example triangle: \(\begin{pmatrix} 1 & 3 & 1 \\ 1 & 1 & 2 \end{pmatrix}\)
The Matrix (\(M\))
The "operator" that changes the coordinates. A \(2 \times 2\) matrix containing the transformation rules.
Each number determines how \(x\) and \(y\) are transformed.
The Image (\(P'\))
The resulting position \(\begin{pmatrix} x' \\ y' \end{pmatrix}\).
Calculated by matrix multiplication:
The Transformation Equation
Expanded form: \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}\)
2. The "Big Four" Standard Transformations
CSEC requires students to memorize or derive specific matrices for four main movements. Use this visual "Cheat Sheet" as your reference.
🪞 Reflection
x-axis: \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
Flips vertically: \(y\) becomes \(-y\)
y-axis: \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
Flips horizontally: \(x\) becomes \(-x\)
Line \(y = x\): \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
Swaps \(x\) and \(y\) coordinates
🔄 Rotation (About Origin)
\(90^\circ\) Anticlockwise: \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)
\((x, y) \rightarrow (-y, x)\)
\(180^\circ\): \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\)
\((x, y) \rightarrow (-x, -y)\)
\(270^\circ\) Anticlockwise: \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)
Same as \(90^\circ\) clockwise
📏 Enlargement
Center \((0,0)\), Scale Factor \(k\):
\(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)
Multiply both coordinates by \(k\)
Identity Matrix:
\(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
The "do nothing" transformation
➡️ Translation
Special Case: Translation does not use matrix multiplication!
\(\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}\)
Vector addition, not matrix multiplication
3. Translation Vectors
Explain that Translation is the only transformation that does not use a \(2 \times 2\) matrix for multiplication. Instead, it uses Vector Addition.
The Translation Rule
Formula: \(\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} + \begin{pmatrix} a \\ b \end{pmatrix}\)
Where \(\begin{pmatrix} a \\ b \end{pmatrix}\) is the translation vector showing how far to move in the x and y directions.
4. Interactive "Transformation Sandbox"
See the Matrix in Action
Objective: Enter numbers in the 2×2 matrix or use preset buttons to see how they transform the "L" shape. Watch the matrix multiplication happen in real-time.
Transformation Matrix \(M\)
Current Transformation
\( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} \)
5. Combined Transformations (Composite)
This is a high-level CSEC objective. If a shape is transformed by Matrix \(A\) and then by Matrix \(B\):
The Order Matters
The combined matrix is \(B \times A\) (not \(A \times B\)).
Logic: You apply \(A\) first (closest to the point), then \(B\). Remember matrix multiplication is not commutative!
Example: If \(T_1 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) (90° rotation) and \(T_2 = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\) (y-axis reflection), then the combined transformation is:
\(T_2 \times T_1 = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) (reflection in \(y = x\))
6. CSEC Exam Mastery Tips
Avoid These Common Mistakes
Column Format
- Always arrange the coordinates of a shape into a large matrix with each column representing a vertex.
- Example: Triangle with vertices (1,1), (3,1), (1,2) becomes \(\begin{pmatrix} 1 & 3 & 1 \\ 1 & 1 & 2 \end{pmatrix}\).
- Transform the whole shape in one multiplication.
Origin Check
- Remember that standard rotation and enlargement matrices only work when the center is the origin \((0,0)\).
- If the center is not the origin, you must translate to origin, transform, then translate back.
- This often requires composite transformations.
The "Unit Square" Trick
- If you forget a matrix, see what happens to points \((1,0)\) and \((0,1)\).
- Their new positions become the first and second columns of your matrix.
- Example: For 90° rotation, (1,0) → (0,1) and (0,1) → (-1,0), giving \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\).
The unit square trick: Transform (1,0) and (0,1) to find the matrix columns
7. Worked Example: Mapping the Image
Problem: A triangle has vertices \(A(1,1), B(3,1),\) and \(C(1,2)\). Find its image under the transformation \(M = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\).
Set up the Matrix Multiplication:
\[ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 3 & 1 \\ 1 & 1 & 2 \end{pmatrix} \]
Arrange original vertices as columns in a \(2 \times 3\) matrix.
Multiply:
New \(x\) values: \((0 \times x) + (-1 \times y) = -y\)
New \(y\) values: \((1 \times x) + (0 \times y) = x\)
Calculate each vertex:
\(A(1,1) \rightarrow A'(-1, 1)\)
\(B(3,1) \rightarrow B'(-1, 3)\)
\(C(1,2) \rightarrow C'(-2, 1)\)
Result in matrix form:
\[ \text{Image Vertices} = \begin{pmatrix} -1 & -1 & -2 \\ 1 & 3 & 1 \end{pmatrix} \]
Identify the Transformation:
This matrix represents a \(90^\circ\) Anticlockwise Rotation about the origin.
Check: \((x, y) \rightarrow (-y, x)\) matches the 90° ACW rule.
8. Practice Mission: "The Secret Code"
Step 1: Reflection in x-axis matrix: \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
Step 2: Enlargement ×2 matrix: \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)
Step 3: Combined transformation: \(\begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}\)
Step 4: Apply to each point: \((x,y) \rightarrow (2x, -2y)\)
\((2,1) \rightarrow (4,-2)\)
\((4,1) \rightarrow (8,-2)\)
\((2,3) \rightarrow (4,-6)\)
\((4,3) \rightarrow (8,-6)\)
\((3,2) \rightarrow (6,-4)\)
Decoded message: These points form the letter "H" when plotted.
Step 1: Reflection in \(y = x\): \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
Step 2: 180° rotation: \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\)
Step 3: Combined (180° rotation THEN reflection in y=x):
Remember order matters! If reflection is first, then rotation:
\(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\)
Final Answer: \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\)
\[ \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} (2 \times 3) + (1 \times 4) \\ (-1 \times 3) + (3 \times 4) \end{pmatrix} = \begin{pmatrix} 6 + 4 \\ -3 + 12 \end{pmatrix} = \begin{pmatrix} 10 \\ 9 \end{pmatrix} \]
Answer: \(P'(10, 9)\)
Transformation Matrix Cheat Sheet
Reflection
x-axis: \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
y-axis: \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
\(y=x\): \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
Rotation (Origin)
90° ACW: \(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)
180°: \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\)
270° ACW: \(\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)
Enlargement
Scale factor \(k\): \(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)
Identity: \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
Composite Order
\(T_2\) then \(T_1\) = \(T_1 \times T_2\)
Apply right to left: \(T_2\) first, then \(T_1\)