Notes on Matrices and Transformations

Introduction

• Lecturer: Vincent Conway
• Topic: Matrices and Transformations

---

Matrix Transformation of Triangle ABC

Given Triangle Vertices:

• A = (5, 6)
• B = (2, 2)
• C = (2, x)

Transformation Matrix:

• Matrix T = [ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} ]

Results of Transformation

• Transformed vertices:
  • A' = (-6, -2)
  • B' = (5, 2)
  • C' = (-2, 2)

Cartesian Plane Visualization

• The size remains the same post-transformation.
• Orientation changes (positive 90 degrees rotation).

---

Identifying the Type of Transformation

• Pre-multiplying the matrix identifies the transformation.
• Stepping through key points to visualize transformations about the origin.
• Applications of perpendicular bisectors in analysis.

---

Finding Matrices of Transformation

Given Image Vertices:

• A' = (-1, 4)
• B' = (-1, 1)
• C' = (-3, 1)

Setup for Transformation Matrix

• Transformation represented as T = [\begin{bmatrix} A & B \ C & D \end{bmatrix}]

Example Calculations:

• System of equations formed:
  1. A + 4B = -1
  2. A + B = -1
  3. 3A + B = -3
  4. C + D = 1

Solving For Variables

• A = -1, B = 0, C = 0, D = 1

---

Successive Transformations

Steps to Calculate

• Apply transformations in order: Perform Transformation Y first, then Transformation X.
• Example: For triangle ABC and transformations X and Y.

Resulting Transverse

• The final image comparison yields new vertices after the successive transformations.

---

Single Matrix Representation

• To combine transformations into a single matrix, multiply them sequentially: [T = T_1 \times T_2]

---

Inverse Transformation

General Method:

1. If given a matrix, find its inverse by ensuring:
   • X * X_inverse = Identity Matrix (I)

---

Area Scale Factors and Determinants

Definitions:

• Area Scale Factor = Area of Image / Area of Object
• Determinant can relate to area transformations via:
  • |det(M)| = Area Scale Factor

Calculating Det(M) Example

1. For a square with matrix
2. Determinant = ad - bc

---

Shear and Stretch Matrices

Definitions:

• Shear Matrices have invariant lines along which points do not vary during transformation.

Example Shear Matrix Representation:

1. Matrices represent invariant lines (e.g., X or Y axes).
2. Describe scaling via identified invariant and affected points.

---

Final Notes

• The discussed matrices and transformations highlight significant geometric manipulations. Understanding these concepts is crucial for further explorations in linear algebra.