IB Mathematics AI AHL Geometric transformations MAI Study Notes- New Syllabus

IB Mathematics AI AHL Geometric transformations MAI Study Notes

LEARNING OBJECTIVE

Geometric transformations of points in two dimensions using matrices:

Key Concepts:

Geometric transformations
Compositions of the transformations.
Determinant of a Transformation Matrix

A transformation matrix is a 2 × 2 matrix that linearly transforms a point $P(x, y)$ in the plane to a new point $P'(x’, y’)$. The transformation is represented using matrix multiplication:

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

is the transformation matrix, and

$
P = \begin{pmatrix} x \\ y \end{pmatrix}
$

is a point, then the transformed point $P’$ is:

$
P’ = M \cdot P = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}
$

So the coordinates of the image point are:

$
x’ = ax + by, \quad y’ = cx + dy
$

This operation shows how a matrix transforms a point using linear combinations of its components.

Example

Let the transformation matrix be

$
M = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
$

Transform the point

$
A = \begin{pmatrix} 1 \\ 1 \end{pmatrix}
$

▶️Answer/Explanation

Solution:

$
M \cdot A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 + 2 \\ 3 + 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}
$

So the point $A(1, 1)$ is mapped to $A'(3, 7)$.

In general, for any point $\begin{pmatrix} x \\ y \end{pmatrix}$, the transformation becomes:

$
\begin{pmatrix} x’ \\ y’ \end{pmatrix} = \begin{pmatrix} x + 2y \\ 3x + 4y \end{pmatrix}
$

Using this matrix, we can find the images of several points:

$O(0, 0) \rightarrow O'(0, 0)$
$A(1, 1) \rightarrow A'(3, 7)$
$B(-1, 2) \rightarrow B'(3, 5)$

This transformation maps shapes as well as individual points.

For example:

The line through $O(0, 0)$ and $A(1, 1)$ is mapped to the line through $O'(0, 0)$ and $A'(3, 7)$.

The triangle $OAB$ with vertices $O(0, 0)$, $A(1, 1)$, and $B(-1, 2)$ is mapped to the triangle $O’A’B’$ with vertices $O'(0, 0)$, $A'(3, 7)$, and $B'(3, 5)$.

BASIC TRANSFORMATION MATRICES
Transformation	Matrix
Horizontal stretch by scale factor $ k $	$ \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} $
Vertical stretch by scale factor $ k $	$ \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} $
Enlargement by scale factor $ k $, center at $ (0, 0) $	$ \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $
Rotation anticlockwise by angle $ \theta $ about origin	$ \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $
Rotation clockwise by angle $ \theta $ about origin	$ \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} $
Reflection in line $ y = mx $, where $ m = \tan \theta $	$ \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} $

Example

The matrix

$ A = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} $

maps

$\begin{pmatrix} x \\ y \end{pmatrix}$ to $\begin{pmatrix} 3x \\ y \end{pmatrix}$.

▶️Answer/Explanation

Solution:

$
\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x \\ y \end{pmatrix}
$
Thus, it stretches each point horizontally with a scale factor of 3. It also stretches any shape horizontally with a scale factor of 3. For example, it maps the triangle $ OAB $ with vertices $ O(0, 0) $, $ A(2, 0) $, $ B(1, 1) $ to the triangle $ O’A’B’ $ with vertices $ O'(0, 0) $, $ A'(6, 0) $, $ B'(3, 1) $.

Example

Consider the triangle $ OAB $ with $ O(0, 0) $, $ A(2, 0) $, $ B(1, 1) $.

The image of the triangle under the following transformations is shown below:

1. Vertical stretch with a scale factor of 2:

2. Enlargement with a scale factor of 2, center $ (0, 0) $:

3. Clockwise rotation of $ 90^\circ $ about the origin:

4. Reflection in the line $ y = x $ ($ m = \tan 45^\circ = 1 $):

▶️Answer/Explanation

Solution:

Consider the triangle $ OAB $ with $ O(0, 0) $, $ A(2, 0) $, $ B(1, 1) $.

The image of the triangle under the following transformations is shown below:

1. Vertical stretch with a scale factor of 2:
$
M = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}
$
Maps:
$
O(0, 0) \mapsto O'(0, 0), \quad A(2, 0) \mapsto A'(2, 0), \quad B(1, 1) \mapsto B'(1, 2)
$

2. Enlargement with a scale factor of 2, center $ (0, 0) $:
$
M = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}
$
Maps:
$
O(0, 0) \mapsto O'(0, 0), \quad A(2, 0) \mapsto A'(4, 0), \quad B(1, 1) \mapsto B'(2, 2)
$

3. Clockwise rotation of $ 90^\circ $ about the origin:
$
M = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
$
Maps:
$
O(0, 0) \mapsto O'(0, 0), \quad A(2, 0) \mapsto A'(0, -2), \quad B(1, 1) \mapsto B'(1, -1)
$

4. Reflection in the line $ y = x $ ($ m = \tan 45^\circ = 1 $):
$
M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}
$
Maps:
$
O(0, 0) \mapsto O'(0, 0), \quad A(2, 0) \mapsto A'(0, 2), \quad B(1, 1) \mapsto B'(1, 1)
$

COMPOSITION OF TRANSFORMATIONS

The composition of transformations refers to applying two or more transformations in sequence. If transformation $A$ is represented by matrix $M_1$, and transformation $B$ is represented by matrix $M_2$, then the combined transformation is represented by the matrix product $M_2 M_1$. This means that $M_1$ is applied first, followed by $M_2$.

In general, if:

$
\text{Point } P \xrightarrow{M_1} P’ \xrightarrow{M_2} P”
$

then the overall transformation is:

$
P \xrightarrow{M_2 M_1} P”
$

If a vector $\begin{pmatrix} x \\ y \end{pmatrix}$ is transformed:

First by matrix $ A $,
Then by matrix $ B $,
Then by matrix $ C $,

we obtain:

$
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto A \begin{pmatrix} x \\ y \end{pmatrix} \mapsto BA \begin{pmatrix} x \\ y \end{pmatrix} \mapsto CBA \begin{pmatrix} x \\ y \end{pmatrix}
$

Hence, the composition of the three transformations can be achieved by the product of the three matrices (in the reverse order).

Example

$\begin{pmatrix} x \\ y \end{pmatrix}$

be transformed by:

1. A horizontal stretch with a scale factor of 3:
2. A clockwise rotation of angle $ \theta = 90^\circ $:

▶️Answer/Explanation

Solution:

1. A horizontal stretch with a scale factor of 3:
$
A = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}
$
2. A clockwise rotation of angle $ \theta = 90^\circ $:
$
B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
$

$
\begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x \\ y \end{pmatrix}
$
and then:
$
\begin{pmatrix} 3x \\ y \end{pmatrix} \mapsto \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 3x \\ y \end{pmatrix} = \begin{pmatrix} y \\ -3x \end{pmatrix}
$

We obtain the same result by applying the transformation matrix:
$
BA = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -3 & 0 \end{pmatrix}
$
Indeed:
$
\begin{pmatrix} 0 & 1 \\ -3 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} y \\ -3x \end{pmatrix}
$

In general, the composition of $ n $ transformations with matrices $ A_1, A_2, \ldots, A_n $ can be achieved by the product of matrices $ A_n \cdots A_2 A_1 $.

AN INTERESTING PROPERTY OF THE DETERMINANT

Suppose that a transformation matrix $ M $ maps a closed shape $ S $ to a new shape $ S’ $. Then:

$
\text{Area of } S’ = |\det M| \times \text{Area of } S
$

Example

Consider again the triangle OAB with O(0,0), A(2,0), B(1,1). , the original triangle $ OAB $ has area 1.

1. Vertical stretch:

2. Enlargement:

3. Clockwise rotation $ 90^\circ $:

4. Reflection in $ y = x $:

▶️Answer/Explanation

Solution:

1. Vertical stretch:
$
M = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}, \quad \det M = 2 \\
\text{Area of image} = 2
$

2. Enlargement:
$
M = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}, \quad \det M = 4 \\
\text{Area of image} = 4
$

3. Clockwise rotation $ 90^\circ $:
$
M = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}, \quad \det M = 1 \\
\text{Area of image} = 1
$

4. Reflection in $ y = x $:
$
M = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \det M = -1 \\
\text{Area of image} = 1
$

Previous Topic Notes

Next Topic Notes

Transformation	Matrix
Horizontal stretch by scale factor \( k \)	\( \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} \)
Vertical stretch by scale factor \( k \)	\( \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} \)
Enlargement by scale factor \( k \), center at \( (0, 0) \)	\( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \)
Rotation anticlockwise by angle \( \theta \) about origin	\( \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \)
Rotation clockwise by angle \( \theta \) about origin	\( \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \)
Reflection in line \( y = mx \), where \( m = \tan \theta \)	\( \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix} \)

IB Mathematics AI AHL Geometric transformations MAI Study Notes- New Syllabus

IB Mathematics AI AHL Geometric transformations MAI Study Notes

TRANSFORMATION MATRICES

BASIC TRANSFORMATION MATRICES

COMPOSITION OF TRANSFORMATIONS

AN INTERESTING PROPERTY OF THE DETERMINANT

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