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Linear maps and projections

A **linear map** transforms vectors in a structured way (stretch, rotate, shear). A **projection** flattens vectors onto a subspace—like casting a shadow.

What it is

A linear map transforms vectors in a structured way (stretch, rotate, shear). A projection flattens vectors onto a subspace—like casting a shadow.

Why it matters

Layers, PCA intuitions, and least squares all lean on linear maps. Pictures beat symbol fear.

How it works (plain)

Matrices encode maps. Projections find the closest point in a simpler space—useful for compression and for understanding “components.”

Everyday example

A sundial shadow is a projection of a 3D stick onto a 2D plane.

Try it

Draw a vector and its shadow on the x-axis—that’s a projection sketch.

Myths

⚠️ Myth: Linear means “simple and weak.”
✓ Reality: High-dimensional linear maps are expressive; nonlinearities stack them into deep nets.

Sources