Linear Transformation of a Unit Square

Scene Overview

• Purpose: Visually demonstrate a linear transformation $T(x)=Ax$ on the 2‑D plane, showing how a unit square at the origin is mapped to a parallelogram, how the grid deforms, and how the standard basis vectors $\hat{i}$ and $\hat{j}$ transform. The original square stays blue, the transformed shape yellow, and the matrix $A$ is displayed numerically.
• Duration: ~20 seconds (well under the 30‑second guideline).

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1. Mathematical Elements

• Transformation equation: $$T(\mathbf{x}) = A\mathbf{x}$$
• Matrix: Display a generic 2×2 matrix with concrete numbers, e.g.
  $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}$$
  (The numbers can be swapped later if the user provides a different matrix.)
• Unit square: Vertices $(0,0), (1,0), (1,1), (0,1)$.
• Basis vectors: $\hat{i}= (1,0)$ and $\hat{j}= (0,1)$.

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2. Visual Elements

Element	Shape / Object	Color	Notes
Coordinate plane	2‑D axes with grid lines	Axes: dark gray; Grid: light gray	Grid extends beyond the unit square to show deformation.
Original unit square	Filled polygon (square)	Blue (fill) with a thin dark‑blue stroke	Centered at the origin.
Transformed shape	Filled polygon (parallelogram)	Yellow (fill) with a thin orange stroke	Starts invisible, fades in as transformation proceeds.
Basis vectors	Arrow from origin to (1,0) and (0,1)	Blue arrows (original) and later orange arrows (transformed)	Arrowheads clearly visible.
Matrix display	Inline LaTeX matrix	White text on a semi‑transparent dark rectangle background	Appears in the top‑right corner throughout the animation.
Transformation equation	Inline LaTeX $T(\mathbf{x}) = A\mathbf{x}$	White text on same background as matrix	Appears briefly at the start, then fades out.

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3. Animation Sequence & Timing

Time (s)	Action
0.0 – 1.5	Fade‑in the coordinate axes, grid, and the blue unit square. Simultaneously display the matrix $A$ (with background) in the top‑right corner.
1.5 – 2.5	Appear the equation $T(\mathbf{x}) = A\mathbf{x}$ below the matrix, then fade it out by 2.5 s.
2.5 – 3.5	Show the two basis arrows $\hat{i}$ and $\hat{j}$ in blue, anchored at the origin.
3.5 – 8.0	Core transformation: Over 4.5 s, smoothly morph the blue square into the yellow parallelogram using the linear map $A$. Simultaneously:

• Deform the background grid (each grid line is transformed by $A$).
• Interpolate the blue basis arrows to orange arrows representing $A\hat{i}$ and $A\hat{j}$.
• Fade the original blue square out while fading the yellow parallelogram in, keeping a brief overlap for visual continuity. |
  | 8.0 – 9.0 | Hold the final configuration (yellow parallelogram, deformed grid, orange basis arrows) for a moment to let the viewer absorb the result. |
  | 9.0 – 10.0 | Fade out the transformed shape and grid, leaving only the original axes and the matrix display for a quick recap. |
  | 10.0 – 12.0 | Optional: Reverse the transformation (yellow → blue) to emphasize invertibility, but only if time permits; otherwise end scene. |
  | End | Fade out everything. |

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4. Camera & Perspective

• Static 2‑D view: Camera remains fixed, centered on the origin. No zoom or pan is required; the entire unit square and its image stay within the frame throughout.
• Scale: Choose a view window of $[-2, 3] \ imes [-2, 3]$ (or similar) to comfortably contain the deformed grid.

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5. Additional Details

• Background: Light off‑white to keep grid lines visible.
• Opacity: Grid lines at 30 % opacity; deformed grid lines retain the same opacity for consistency.
• Transition easing: Use smooth (quadratic‑in‑out) interpolation for the morph and grid deformation to convey a natural linear mapping.
• Text background: The matrix and equation text sit on a semi‑transparent dark rectangle (≈70 % opacity) to guarantee readability over any underlying graphics.
• No extra text: Apart from the matrix/equation, no additional labels are used; the visual change itself conveys the concept.

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6. Summary

The scene starts with a clear, static coordinate system and a blue unit square. The matrix $A$ is shown, then the square, grid, and basis vectors smoothly transform under $A$ into a yellow parallelogram, a deformed grid, and orange basis arrows, illustrating the linear map $T(x)=Ax$. The whole animation fits comfortably within ~20 seconds, uses a single Manim Scene class, and adheres to all given constraints.