Question

I need help with 2 of the 3 exercises or with the 3 exercises.

LINEAR ALGEBRA TOPICS: Quadratic Forms and Sylvester's Theorem May 23, 2019 1.Let V be a real vector space of finite dimension and f: VR a function such that the expression F(v, w)-f(v+w)- f(v)-f(w) is bilinear. Assume further that f(λυ-λ2f(v) is satisfied for all λ E R and every vector UEV Prove that under these conditions f is in fact a quadratic form. Determine the bilinear form associated with f and show that the bilinear form is unique. 2.Determine the nullity index and the positivity index for the determined internal products, for each of the following matrices 3·Let V be a real vector space of finite dimension and B : V × V → R a scalar product. Prove that V admits a decomposition in direct sum of the form: Where:

Answer 1

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