Projecting onto Subspaces

Linear Algebra (Matrices) · Axiom Academy

LESSON Projecting onto a Subspace Projecting onto a line was a warm-up. With an orthogonal basis, the same idea lands a vector on a whole plane. Let W be a plane through the origin, and a vector poking out of it. Drop straight down onto W . Where it lands — the foot of the perpendicular — is the projection . What's left over, , stands perpendicular to the whole plane. 2. Build It From an Orthogonal Basis How do we actually compute ? Take an orthogonal basis of W (perpendicular vectors). Project onto each one separately — just the single-line formula — then add those pieces tip‑to‑tail. Because the are mutually perpendicular, the pieces don't interfere, and their sum is exactly . one single-line projection per basis vector, summed Worked numbers — take (note , orthogonal) and : 3. It's the Closest Point in W Why does deserve to be called the projection? Slide a test point all around the plane and measure . That distance bottoms out at exactly one spot — the foot . Every other point of W is strictly farther from . So is the best approximation to from within W . : the gap is as small as it can be — here . Move off the foot and the distance only grows — the gap tilts and lengthens. Projecting onto a subspace is the same shadow idea as projecting onto a line — an orthogonal basis just lets you do it one perpendicular direction at a time. Scroll up to revisit any step.

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