Matrix Transformations in 2D

Description

Visualizes how 2x2 matrices transform the plane by animating a NumberPlane grid through four key transformations: identity, 90-degree rotation, shear, and stretch. The basis vectors i-hat and j-hat are highlighted to show how the matrix columns determine the transformation outcome.

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Phases

#	Phase Name	Duration	Description
1	Introduction	5s	Display title, show NumberPlane with labeled basis vectors i-hat and j-hat
2	Identity Matrix	6s	Show identity matrix MathTex, animate (no change), explain columns = basis images
3	Rotation 90°	8s	Show rotation matrix, animate grid rotating 90°, basis vectors trace new positions
4	Shear Transform	8s	Show shear matrix, animate horizontal shear, vertical lines tilt
5	Stretch Transform	8s	Show stretch/scale matrix, animate axes-aligned scaling
6	Composition	10s	Show two transforms composing, matrix multiplication = sequential transforms
7	Summary	5s	Display all four matrices side by side with their geometric interpretations

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Layout

+--------------------------------------------------+
|  Title: "Matrix Transformations in 2D"           |
+--------------------------------------------------+
|                                                  |
|   NumberPlane (center-left)  | Matrix MathTex   |
|   with grid lines            | (center-right)   |
|                              |                  |
|   i-hat (red arrow)          | [a  b]           |
|   j-hat (green arrow)        | [c  d]           |
|                              |                  |
|   Transformed grid           | Equation label   |
|   (animates via ApplyMatrix) |                  |
|                              |                  |
+--------------------------------------------------+
|  Bottom: formula / description text             |
+--------------------------------------------------+

Area Descriptions

• Left 60%: NumberPlane with animated grid transformation
• Right 40%: Matrix display (MathTex), transformation name, and formula
• Bottom strip: Descriptive text about the current transformation

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• Use ApplyMatrix for grid transformation animations
• Basis vectors should be drawn with Arrow objects and labeled with MathTex(r"\hat{i}") and MathTex(r"\hat{j}")
• Reset the plane between each transformation using Transform back to identity
• Color scheme: i-hat = RED, j-hat = GREEN, grid = BLUE tones, transformed = YELLOW