Orthogonality and Least Squares Summary

Linear Algebra (Matrices) · Axiom Academy

SUMMARY Unit Summary: Orthogonality and Least Squares A whole-unit review: how perpendicularity — orthogonal sets, Gram-Schmidt, projections, and the normal equations — turns hard linear-algebra problems into clean computations. Two vectors are orthogonal exactly when their dot product is zero: — the algebraic test for the geometric idea of "perpendicular" in any . An orthonormal basis makes coordinates effortless: just take dot products, — no linear system to solve. Gram-Schmidt turns any basis into an orthogonal one by projecting and subtracting; normalizing gives orthonormal vectors and the factorization A=QR . The orthogonal projection of onto a subspace W is the closest point in W , and the residual is orthogonal to W . That projection is exactly least squares : the best-fit solution of an inconsistent solves the normal equations . Core Concept Dot Product & Orthogonality The dot product measures alignment; it is zero precisely when the vectors are perpendicular. It also gives length, , extending geometry from to all of . When to use: testing perpendicularity, computing lengths and angles. Watch out for: orthogonal is about the dot product, not about looking "at right angles" on a picture. Core Concept Orthogonal & Orthonormal Sets A set is orthogonal if every pair has dot product 0 ; orthonormal adds unit length. A key fact: an orthogonal set of nonzero vectors is automatically linearly independent , so it is a basis for its span.

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