This sandbox was generated by Claude (Anthropic) for the Linear Algebra 101 course materials, and is built for Module 6 (Inner Product Spaces). Pick a periodic target — square, sawtooth, triangle or pulse — and watch the page compute its Fourier coefficients aₖ, bₖ as inner products of the function with each sine and cosine, then rebuild the function from the first N harmonics. Every drawn point is computed — the reconstruction is the sum of the harmonics, each coefficient bar is an integral, and the error curve is measured, never sketched.

The Fourier series is a projection. The sines and cosines are orthogonal under the inner product ⟨f, g⟩ = ∫ f g dt over one period, so the best coefficient of each is simply the inner product of f with it — the same move as projecting a vector onto an orthonormal basis in Rⁿ. No optimisation is run: the integral is the answer. The page prints the orthogonality table so you can watch the off‑diagonal inner products sit at zero. At a jump, the partial sum overshoots by about 9% of the jump — the Gibbs phenomenon — and that overshoot narrows but never shrinks as N grows.

Notation follows the course: period T, fundamental frequency f₀ = 1/T, cosine coefficients aₖ, sine coefficients bₖ, DC term a₀/2, inner product ⟨f, g⟩ = ∫ f g dt over one period, and N harmonics kept.

What is chosen rather than computed: the default waveform, the preset waveforms, the drawn window (two periods) and the quadrature resolution. The Gibbs constant (~8.95% of the jump) is a published fact, labelled where it appears — and the page also measures its own overshoot rather than printing it.