Mr. Integral

THE BIG IDEA

Orthogonality and Least Squares

What is the closest answer when an exact fit is impossible?

Projection separates a vector into a part you can represent and a perpendicular residual. Build an orthonormal basis to calculate that split efficiently, then extend it to fitting inconsistent data.

Begin the first lesson →See the lesson route ↓

3 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Projection
  2. computing projections efficiently uses02Orthonormal coordinates
  3. the closest column-space vector solves03Least squares

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Lessons in this unit3 lessons
  1. 01Dot Products and Orthogonal ProjectionSeparate a vector into its component along a direction and a perpendicular residual.
  2. 02Gram–Schmidt and QR FactorizationConstruct an orthonormal basis and use triangular equations to recover coordinates.
  3. 03Least Squares and Data FittingFit an inconsistent system by minimizing residual length and checking its orthogonality.