(a) Define the quadratic form qÃ(x₁,,xn) associated to a real symmetric n × n matrix A. (b) Show that if two real symmetric matrices A, B are congruent then there is a linear change of variables to x₁,...,xn such that ¶₁(x₁, ,xn) = 9B (x₁, ,xn). (c) Show that the following quadratic form on R³ is not positive definite: q(x, y, z) = x² + 4xz + 3y² + 4yz + z². ... (d) Let V be a real vector space with basis v₁, v2 and define a dot product by V₁ V₁ = 1, V₁ V2 V₂2 V₁ = λ₁ . V₂ V2 = 2, where A E R is fixed. For what values of A does (V,.) become an inner product space with the stated dot products? (Hint: you may wish to diagonalise the associated quadratic form q(x, y) = x² + 2xXy + 2y².)
(a) Define the quadratic form qÃ(x₁,,xn) associated to a real symmetric n × n matrix A. (b) Show that if two real symmetric matrices A, B are congruent then there is a linear change of variables to x₁,...,xn such that ¶₁(x₁, ,xn) = 9B (x₁, ,xn). (c) Show that the following quadratic form on R³ is not positive definite: q(x, y, z) = x² + 4xz + 3y² + 4yz + z². ... (d) Let V be a real vector space with basis v₁, v2 and define a dot product by V₁ V₁ = 1, V₁ V2 V₂2 V₁ = λ₁ . V₂ V2 = 2, where A E R is fixed. For what values of A does (V,.) become an inner product space with the stated dot products? (Hint: you may wish to diagonalise the associated quadratic form q(x, y) = x² + 2xXy + 2y².)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 34E
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Let V be a real vector space with basis v1, v2 and define a dot product by
v1 · v1 = 1, v1 · v2 = v2 · v1 = λ, v2 · v2 = 2,
where λ ∈ R is fixed. For what values of λ does (V, ·) become an inner product
space with the stated dot products? [7]
(Hint: you may wish to diagonalise the associated quadratic form
q(x, y) = x
2 + 2xλy + 2y
2
.)
Solution
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