article = "https://physicslibrary.org/encyclopedia/InnerProductsOfFunctions.html"
article_alias = "FS02"
article_id = 1519
julia_version = "1.10.12"
reference_generator = "julia --startup-file=no --project=. inner-product-lab.jl"
reference_method = "exact rational polynomial moments and finite sums; independent of browser floating-point evaluation"
browser_method = "analytic polynomial-product integration, Euclidean and Frobenius sums; separate midpoint approximations"
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scope = "real dimensionless vectors in R2, degree-at-most-three polynomial function spaces, and real 2x2 matrices with the Frobenius metric"
projection = "c*=inner(u,v)/inner(v,v); r=u-c*v; E(c)=inner(r,r); trial and optimum remain distinct"
angle = "acos(inner(u,v)/(norm(u)*norm(v))) for nonzero objects; abstract span diagram is an isometric representation, not the angle of function graphs"
weight = "uniform 1, right ramp 2(x-a)/(b-a), or 1+x^2; ramp is positive in the interior with a zero endpoint"
sampling = "4 to 128 midpoint rectangles include w(x_i)*dx; analytic scalar and approximate sample sum are reported separately"
matrix = "inner(A,B)=trace(transpose(A)*B)=sum(A_ij*B_ij); identity projection extracts half the trace"
normalization = "target rescaled to unit norm; trial coefficient adjusted to preserve the trial projection; zero target cannot be normalized"
zero_cases = "angle undefined if either object is zero; zero target has a zero projection object, non-unique coefficient and constant squared error"
graphics = "source and error axes adapt with visible numerical ticks; spatial and abstract vector-plane axes retain equal units; matrix heatmaps share a scale"
precision = "17-digit CSV references; screen readouts suppress magnitudes below 1e-10; orthogonality checks are numerical, not symbolic claims"
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