# Concept derivation guides

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Concept guide



## [Real bilinear form: proving bilinearity slot by slot from x^T A y, with exact evaluation](/en/derivations/derivation-mko-real-bilinear-form-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-real-bilinear-form)

Let A=[[1,2],[−1,3]] and B(x,y)=x^T A y for x, y∈ℝ². Prove separately that B is linear in the first slot and in the second slot (bilinear); evaluate exactly B(x,y)=7 and B(y,x)=−20 for x=(2,−1), y=(1,4), showing that bilinearity does not require symmetry; and prove that the black-box function f(x,y)=x1y1+(x2³−x2)y2 passes some additivity samples but is not bilinear.

Real bilinear form

derivation-mko-real-bilinear-form-01





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