(a) Define the quadratic form qA(x1, ,xn) associated to a real symmetric n × nmatrix A.(b) Show that if two real symmetric matrices A, B are congruent then there is alinear change of variables to x₁,, such that qA (X₁, ···,xn) = ¶B(x₁, ···, x').9(c) Show that the following quadratic form on R³ is not positive definite:q(x, y, z) = x² + 4xz + 3y² + 4yz + z².xn(d) Let V be a real vector space with basis V₁, V2 and define a dot product byV1 V1 1,V2 V1 = 1, V2 V2 ==V1 V2=●●2,where A E R is fixed. For what values of X does (V, .) become an inner productspace with the stated dot products?(Hint: you may wish to diagonalise the associated quadratic formq(x, y) = x² + 2xy + 2y².)

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Answered: (a) Define the quadratic form…

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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Question

(a) Define the quadratic form qA(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,, such that qA (X₁, ···,xn) = ¶B(x₁, ···, x').
9
(c) Show that the following quadratic form on R³ is not positive definite:
q(x, y, z) = x² + 4xz + 3y² + 4yz + z².
x
n
(d) Let V be a real vector space with basis V₁, V2 and define a dot product by
V1 V1 1,
V2 V1 = 1, V2 V2 =
=
V1 V2
=
●
●
2,
where A E R is fixed. For what values of X 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 + 2xy + 2y².)

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Follow-up Question

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
.)

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