Lesson Notes By Weeks and Term v4 - SHS 3

Lesson Video

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Performance objectives

• Define linear transformations and their matrix representation.
• Calculate the image of a point under a transformation.
• Describe the geometric effect of a transformation.
• Apply transformations to solve problems.

Lesson summary

This lesson explores one of the most powerful applications of algebra: using matrices to describe geometric transformations. We see transformations all around us – when we rotate a picture on our phone, when an animator makes a character move on screen, or even in the symmetrical patterns of our beautiful Kente cloth and Adinkra symbols. By the end of this lesson, you will understand how a simple 2x2 matrix can be used to model complex movements like reflections, rotations, and shears on a 2D plane. This bridges the gap between abstract algebra (matrices) and visual geometry (shapes and points).

Lesson notes

A. What is a Linear Transformation? A linear transformation is a rule that moves or maps every point on a Cartesian plane to a new position. It is "linear" because it has two key properties: It maps straight lines to other straight lines. The origin (0, 0) remains fixed in its position.

In SHS Additional Mathematics, we represent these transformations using 2x2 matrices. B. Representing Points and Transformations Point (Object): A point P with coordinates (x, y) is represented as a 2x1 column vector or position vector, p = $\begin{pmatrix} x \\ y \end{pmatrix}$. Transformation Matrix: The rule for the transformation is represented by a 2x2 matrix, M = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$. Point (Image): The new point P' with coordinates (x', y') is also represented as a column vector, p' = $\begin{pmatrix} x' \\ y' \end{pmatrix}$. C. Finding the Image of a Point (Object → Image) To find the image of a point under a transformation, we pre-multiply the position vector of the point by the transformation matrix.

Formula: Image = Matrix × Object $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}$

Evaluation guide

• Use linear transformation to determine image and object points.
• Think -pair- share, Talk for Learning, and Building on what others say.
• determine image and object points.
• Image of a point under linear transformation