Lesson Notes By Weeks and Term v4 - SHS 3
APPLICATIONS OF ALGEBRA
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APPLICATIONS OF ALGEBRA
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Subject: Additional Mathematics
Class: SHS 3
Term: 1st Term
Week: 12
Grade code: 3.1.2.LI.3
Strand code: 1
Sub-strand code: 2
Content standard code: 3.1.2.CS.2
Indicator code: 3.1.2.LI.3
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATIONS OF ALGEBRA
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This lesson explores the concept of composite linear transformations. In many real-world applications, an object or point undergoes several transformations one after the other. For instance, in computer animation, a character might be rotated and then moved to a new position. In graphic design, a logo might be enlarged and then reflected. Instead of performing these steps separately, we can combine them into a single, more efficient transformation. This lesson teaches us how to find the single matrix that represents a sequence of transformations. This skill is fundamental in fields like computer graphics, robotics, and engineering design, allowing us to model complex movements algebraically.
2.1. What is a Linear Transformation? (Recap) A linear transformation maps points or vectors in a plane to new positions. In SHS Additional Mathematics, we represent these transformations using 2x2 matrices. A point `P(x, y)` is represented by a position vector `p = [x; y]`. A transformation `T` represented by matrix `M = [a b; c d]` maps the point `P` to its image `P'(x', y')` as follows:
`M * p = p'` `[a b; c d] * [x; y] = [ax + by; cx + dy] = [x'; y']` 2.2. What is a Composite Transformation? A composite transformation is the result of applying two or more linear transformations in a specific sequence.
Imagine you have two transformations: Transformation A, represented by matrix A. Transformation B, represented by matrix B.
If we want to apply transformation A first, and then apply transformation B to the result, this is a composite transformation. We can write this as `B ○ A` (read as "B composed with A") or simply `BA`.