Welcome to Transformations, Matrices, and Vectors!
Hello! This chapter is where Algebra meets Geometry in a fantastic way. You're going to learn how to describe movement, shape changes, and location using powerful mathematical tools: Vectors and Matrices. Don't worry if these terms sound complicated; we will break them down step-by-step. By the end, you'll be able to tell a shape exactly where to go and how to change!
Why is this important? These concepts are fundamental in areas like computer graphics, aviation, physics, and engineering. You are learning the mathematics that makes video games look realistic and helps airplanes navigate!
Section 1: Revisiting 2D Transformations (The Foundation)
Before introducing matrices and vectors, let’s quickly recall the four standard transformations we use to move shapes on a 2D plane.
Quick Review: Describing Transformations Fully
- Translation: Requires the vector, e.g., \(\binom{2}{-3}\).
- Reflection: Requires the line of reflection, e.g., \(y=x\).
- Rotation: Requires the centre of rotation, angle, and direction (Clockwise/Anti-clockwise).
- Enlargement: Requires the centre of enlargement and the scale factor.
Did you know? Translation is the only transformation that moves a shape without changing its orientation or size. It's just a pure slide!
Key Takeaway for Transformations
To fully describe a transformation, you need precise coordinates or equations. We will now learn the tools (vectors and matrices) that give us this precision.
Section 2: Understanding Vectors (Instructions for Movement)
In simple terms, a vector is a quantity that has both magnitude (size/length) and direction. Think of it like a set of driving instructions.
1. Column Vectors
We usually write 2D vectors in the form of a column vector:
$$ \mathbf{a} = \binom{x}{y} $$
- \(x\) is the movement in the horizontal direction (positive is right, negative is left).
- \(y\) is the movement in the vertical direction (positive is up, negative is down).
Example: Position and Displacement Vectors
A point P has coordinates \((4, 1)\). Its position vector (starting from the origin O) is \(\vec{OP} = \binom{4}{1}\).
If you move from point A \((1, 5)\) to point B \((3, 2)\), the displacement vector \(\vec{AB}\) is calculated by \(\text{B} - \text{A}\):
$$ \vec{AB} = \binom{3-1}{2-5} = \binom{2}{-3} $$Analogy: Position vector tells you "where you are" from a fixed starting point (the Origin). Displacement vector tells you "how you got there" from one point to another.
2. Vector Operations
i. Vector Addition and Subtraction
Adding vectors is like following one instruction immediately after another. We add or subtract the corresponding components.
If \(\mathbf{a} = \binom{4}{1}\) and \(\mathbf{b} = \binom{-2}{3}\):
$$ \mathbf{a} + \mathbf{b} = \binom{4+(-2)}{1+3} = \binom{2}{4} $$ii. Scalar Multiplication
Multiplying a vector by a normal number (a scalar) changes its magnitude (length) but not its direction (unless the scalar is negative).
Using \(\mathbf{a} = \binom{4}{1}\):
$$ 3\mathbf{a} = 3\binom{4}{1} = \binom{3 \times 4}{3 \times 1} = \binom{12}{3} $$Vectors \(\mathbf{a}\) and \(3\mathbf{a}\) are parallel because they are scalar multiples of each other.
iii. Finding the Magnitude (Length)
The magnitude (length) of a vector \(\mathbf{a} = \binom{x}{y}\) is written as \(|\mathbf{a}|\). Since the components form a right-angled triangle, we use Pythagoras' theorem:
$$ |\mathbf{a}| = \sqrt{x^2 + y^2} $$Step-by-Step Example: Find the magnitude of \(\mathbf{v} = \binom{5}{-12}\).
- Square the components: \(5^2 = 25\), \((-12)^2 = 144\).
- Add them: \(25 + 144 = 169\).
- Take the square root: \(\sqrt{169} = 13\).
- So, \(|\mathbf{v}| = 13\).
Key Takeaway for Vectors
Vectors describe movement and position. We can add, subtract, multiply by a scalar, and calculate their length using Pythagoras' Theorem.
Section 3: Introducing Matrices (The Calculation Machine)
A matrix is simply a rectangular arrangement of numbers. In this syllabus, we focus mainly on 2x2 matrices (2 rows and 2 columns).
$$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$1. Matrix Operations
i. Addition and Subtraction
This is the easiest operation! You just add or subtract the numbers in the same position (element-by-element).
$$ \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} $$ii. Matrix Multiplication (The Tricky Part!)
You multiply rows by columns. This is the single most important rule in this topic. You must use the Row by Column (RC) method.
Step-by-Step: Multiplying a 2x2 Matrix by a 2x1 Vector
This is how we apply a transformation to a point \((x, y)\):
$$ T \times P = P' \quad \rightarrow \quad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} $$- Top Element (New X): Multiply the Top Row by the Column. \((a \times x) + (b \times y)\)
- Bottom Element (New Y): Multiply the Bottom Row by the Column. \((c \times x) + (d \times y)\)
Example: Apply transformation \(T = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\) to point \((4, 5)\).
$$ \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 4 \\ 5 \end{pmatrix} = \begin{pmatrix} (2 \times 4) + (1 \times 5) \\ (0 \times 4) + (3 \times 5) \end{pmatrix} = \begin{pmatrix} 8+5 \\ 0+15 \end{pmatrix} = \begin{pmatrix} 13 \\ 15 \end{pmatrix} $$The new point is \((13, 15)\).
Common Pitfall: The order matters! \(A \times B\) is usually not the same as \(B \times A\).
iii. The Identity Matrix (\(I\))
This is the "do-nothing" matrix. When you multiply any matrix (or vector) by \(I\), it stays the same.
$$ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$2. Determinants and Inverse Matrices
The determinant of a matrix tells us how the area of a shape changes after the transformation. It is denoted as \(\det(A)\) or \(|A|\).
For matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant is:
$$ \det(A) = ad - bc $$Memory Aid: Think of the main diagonal \((ad)\) minus the reverse diagonal \((bc)\).
The Inverse Matrix (\(A^{-1}\))
The inverse matrix is essential because it undoes the original transformation. If matrix A turns P into P', then \(A^{-1}\) turns P' back into P.
$$ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$Step-by-Step Finding the Inverse:
- Find the determinant \(D = ad - bc\).
- Swap the elements on the main diagonal (\(a\) and \(d\)).
- Change the sign of the other two elements (\(b\) and \(c\)).
- Multiply the new matrix by \(\frac{1}{D}\).
Important Rule: If the determinant is zero (\(ad-bc = 0\)), the matrix is called a singular matrix, and it has no inverse. This means the transformation maps the area down to zero (e.g., squashing a 2D shape onto a line).
Key Takeaway for Matrices
Matrices are tools for calculations and transformations. The determinant and the inverse are critical for understanding how transformations change area and how to reverse them.
Section 4: Matrices Representing Transformations (Focus on Origin)
A 2x2 matrix describes a transformation that keeps the origin \((0, 0)\) fixed.
The Golden Rule: To find the 2x2 matrix (T) for any transformation about the origin, see where the unit vectors \(i = \binom{1}{0}\) and \(j = \binom{0}{1}\) land.
$$ T = \begin{pmatrix} \text{New } i & \text{New } j \end{pmatrix} $$1. Standard Transformation Matrices (About the Origin)
a) Rotation (Anti-Clockwise about Origin)
Anti-clockwise is the standard positive direction.
- 90° ACW: $$ \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$
- 180° (ACW or CW): $$ \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} $$
- 270° ACW (or 90° CW): $$ \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} $$
b) Reflection
- Reflection in the x-axis (\(y=0\)): $$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
- Reflection in the y-axis (\(x=0\)): $$ \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $$
- Reflection in the line \(y=x\): $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $$
c) Enlargement (Centre at Origin, Scale Factor \(k\))
Every point is multiplied by the scale factor \(k\).
$$ \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $$2. Combined Transformations
If you perform transformation A followed by transformation B, the resulting combined matrix (C) is found by multiplying the matrices in reverse order: \(C = B \times A\).
Mnemonic: Transformations happen from right-to-left. The matrix closest to the coordinates acts first.
Example: A shape is reflected in the x-axis (A) and then rotated 90° ACW (B).
$$ C = B \times A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \times \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$The resulting matrix C represents the single transformation equivalent to A followed by B.
Crucial Fact: The determinant of the combined matrix \(C\) is equal to the product of the individual determinants: \(\det(C) = \det(B) \times \det(A)\).
Key Takeaway for Matrices and Transformations
A 2x2 matrix defines a transformation around the origin. We use matrix multiplication to find the new coordinates or to combine multiple transformations.
Quick Review: Transformations, Matrices, and Vectors
You have covered the geometry of movement (transformations) and the algebra used to calculate them (vectors and matrices). Keep practicing the multiplication rules—they are the most vital skill in this chapter!
Summary Checklist:
- Can I add and subtract vectors?
- Can I find the magnitude of a vector?
- Can I multiply 2x2 matrices (Row by Column)?
- Can I calculate the determinant of a 2x2 matrix?
- Can I find the inverse of a 2x2 matrix?
- Do I know the standard matrices for rotation, reflection, and enlargement about the origin?
You've done the hard part—understanding the rules. Now, enjoy applying these powerful tools to geometry problems!