Building Orthogonal Sets

Linear Algebra (Matrices) · Axiom Academy

Why we crave perpendicular axes: with an orthogonal basis, every coordinate reads off on its own — no system to solve. The payoff of perpendicular axes Every vector can be written in terms of a basis — but how hard that is depends entirely on the basis you pick. With a generic (skew) basis, the coordinates are tangled together and you must solve a linear system to find them. With an orthogonal basis, that knot dissolves: each coordinate becomes an independent projection you can read off one axis at a time. Watch a target vector get decomposed along an orthogonal basis . Each coordinate appears by dropping a perpendicular onto one axis, computed alone as — and the two projections snap back together to rebuild exactly. Each coordinate is found by itself — no equations linking them. That independence is the whole reason orthogonal bases are worth building. A skew basis: the coordinates are tangled Here the two axes and are not perpendicular ( ). Drag the target point and watch: there is no way to read one coordinate off a single axis — moving the point changes both coordinates at once, because finding them means solving the system . Notice the coupling: nudge the target and both c_1 and c_2 move. The coordinates aren't independent — they're locked together by the system. An orthogonal basis: the coordinates decouple

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