Linear Algebra · Chapter 10

Orthogonality and least squares

Lengths and angles from the dot product and from inner products in general: orthogonal bases and matrices, orthogonal complements, projections, the Gram–Schmidt process, and least squares for systems that have no solution.

u \cdot v = i \sum u_{i} v_{i}

⟨ u, v ⟩

Q^{T} Q = I

u ⊥ v

∥ v ∥^{2} = ⟨ v, v ⟩

proj_{u} v = \frac{v \cdot u}{u \cdot u} u

A^{T} A x^= A^{T} b

W^{⊥}

Lessons

Build the foundation

06The Gram–Schmidt processAny basis can be straightened into an orthogonal one. Keep the first vector; from the second, remove its projection onto the first; from the third, remove its projections onto the two already made; and so on. What is left each time is at right angles to everything before it.Read lesson →

07Least squaresMeasurements disagree, and a system built from them usually has no solution. Least squares asks for the next best thing: the choice of unknowns that makes the errors as small as possible, measured by the sum of their squares. Geometrically it is a projection, and algebraically one system of equations.Read lesson →

08Chapter practiceBring the chapter together: lengths and angles, orthogonal bases and matrices, complements, inner products, projections, Gram–Schmidt and least squares. No worked example sits above the answers.Read lesson →

Mapping the lesson.

Orthogonality and least squares

Build the foundation