Invitation to Another Dimension

Abstraction: Interactive visual progression from linear functions to matrices and vector fields

Key points:

• Linear functions use only scaling (multiplication) and translation (addition); progression builds from 1-in/1-out up through 2-in/2-out matrix form
• Off-diagonal matrix entries create "crosstalk" (shear) between inputs and outputs — an emergent behavior invisible when studying components in isolation
• A 2D affine transform matrix is 3x3 (homogeneous); points padded with 1 receive translation, vectors padded with 0 do not — same matrix handles both
• Vector fields visualize 2x2 matrix functions as arrows at sampled points; linear maps convert any grid of squares into a tiled parallelogram grid
• Sampling is used to represent higher-dimensional continuous functions; linear transformations stretch the number line consistently across all samples
• The full space of 3D affine matrices is 12-dimensional, giving many independent "knobs" to diagnose and fix transformations

Connections: Linear Algebra · Matrix Operations · Vector Fields · Affine Transformations

Source: https://maxgoldste.in/itad/