The Gram-Schmidt Algorithm

Linear Algebra (Matrices) · Axiom Academy

LESSON The Gram-Schmidt Algorithm Turn any basis into an orthogonal one by subtracting off the part each vector already shares — then scale to unit length. 1. Build : subtract the shadow Keep the first vector untouched: . The second vector leans partly along . Project it onto to find that shared part — its shadow — then subtract the shadow. What is left, , points in the brand-new direction perpendicular to . subtract it → the perpendicular remainder 2. Build : subtract both shadows Now there are two finished vectors, and , which span a plane. The third vector has a shadow on that whole plane — and that shadow is just the sum of its projections onto and onto . Subtract both, and the leftover stands perpendicular to the entire plane : the genuinely new third direction. 3. Normalize: scale each to unit length The vectors are perpendicular but have different lengths. One last step makes the basis orthonormal : divide each by its own length so it becomes a unit vector. Orthogonality is untouched — only the lengths change. Lengths, then the unit vectors You ran Gram-Schmidt end to end: subtract shared shadows to orthogonalize, then rescale to get an orthonormal basis for the same span. Scroll up to revisit any step.

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