1.1 Problem statement and historical remarks For finite dimensional R-vector spaces U and V we consider a symmetric bilinear map B: U×U → V. This then defines a quadratic map QB: U → V by QB(u) = B(u,u). Corresponding to each λ ∈ V ∗ is a R-valued quadratic form λQB on U defined by λQB(u) = λ·QB(u). B is definite if there exists λ ∈ V ∗ so that λQB is positive-definite. B is indefinite if for each λ ∈ V ∗ \ ann(image(QB)), λQB is neither positive nor negative-semidefinite, where ann denotes the annihilator. Given a symmetric bilinear map B: U × U → V, the problems we consider are as follows. 1. Find necessary and sufficient conditions characterizing when QB is surjective. 1 2 Vector-valued quadratic forms 2. If QB is surjective and v ∈ V, design an algorithm to find a point u
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