Orthogonal and Orthonormal Sets

Linear Algebra (Matrices) · Axiom Academy

LESSON Orthogonal & Orthonormal Sets A set of mutually perpendicular vectors is automatically independent — and it turns finding coordinates into a single dot product. 1. What Makes a Set Orthogonal An orthogonal set is a collection of nonzero vectors that are pairwise orthogonal — every distinct pair has dot product zero. Watch the basis assemble: each pair lights up with its mark as the dot product lands on 0 . Orthogonal: every distinct pair is perpendicular Orthonormal: also each has unit length 2. Orthogonal Forces Independence Why is an orthogonal set automatically linearly independent? Suppose some combination is the zero vector. Dot both sides with one vector : every term whose vector is perpendicular to contributes nothing — its shadow on the -axis has length 0 . Watch those cross-terms collapse, leaving a single survivor. 3. Reading Coordinates Straight Off Here is the payoff. To write in an orthogonal basis you do not solve a system — each axis acts on its own. Project onto each independently; the coefficient is . Watch the three components peel off and then reassemble tip-to-tail into . You've seen what makes a set orthogonal, why that guarantees independence, and how it turns coordinates into one dot product per axis. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.