Fig 1QR decompositionHover to advance Gram–Schmidt · keys 1–3 for stagesstage 1 · R₁₁ → 0
Q tips and R needles: orthonormal directions and triangular lengthsA = [[2, 1], [2, 3]] (columns a₁, a₂)A Gram–Schmidt squaring jig. Column 1 locks onto the Q₁ rail and dial R₁₁ reads ‖a₁‖ = 2√2 ≈ 2.83. Column 2 is peeled into a projection along Q₁ (R₁₂ = 2√2 ≈ 2.83) and a residual swung onto Q₂ (R₂₂ = √2 ≈ 1.41). The lit tips are the columns of Q; the three dials are R.