Consider the real quadratic form q : R' → R, defined via:q(x1, x2, X3) = x + 25x; + 125x + 10x1x2 + 20xjx3 + 100x2x3!!(a) Write down a symmetric matrix, [q], representing q in terms of the following ordered basis of R' over R:=(b) Determine C, the real canonical form of q.(c) Determine a real invertible matrix M such that M'[q] M = C.Hence, determine a basis, of R’over R, that is orthogonal with respect to (the symmetric bilinear formassociated to) q. (d) Consider the real quadratic form q': R R, defined, for a real number a, via:q'(x1, x2, x3) = x + ax + 5ax + 10x1x2 + 20x1x3 + 100x2x3Find the set of values of a for which the symmetric bilinear form associated to g' defines a real inner product onR.

Consider the real quadratic form q : R’ → R, defined via: q(x1, x2, X3) = x + 25x; + 125x + 10x1x2 + 20x1x3 + 100x2x3 !3! (a) Write down a symmetric matrix, [q]%, representing q in terms of the following ordered basis of R' over R: --900 E = (b) Determine C, the real canonical form of q.

Consider the real quadratic form q : R’ → R, defined via: q(x1, x2, X3) = x + 25x; + 125x + 10x1x2 + 20x1x3 + 100x2x3 !3! (a) Write down a symmetric matrix, [q]%, representing q in terms of the following ordered basis of R' over R: --900 E = (b) Determine C, the real canonical form of q.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter10: Matrices

Section10.CR: Chapter 10 Review

Problem 15CR

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Question

Consider the real quadratic form q : R' → R, defined via:
q(x1, x2, X3) = x + 25x; + 125x + 10x1x2 + 20xjx3 + 100x2x3
!!
(a) Write down a symmetric matrix, [q], representing q in terms of the following ordered basis of R' over R:
=
(b) Determine C, the real canonical form of q.
(c) Determine a real invertible matrix M such that M'[q] M = C.
Hence, determine a basis, of R’over R, that is orthogonal with respect to (the symmetric bilinear form
associated to) q.

(d) Consider the real quadratic form q': R R, defined, for a real number a, via:
q'(x1, x2, x3) = x + ax + 5ax + 10x1x2 + 20x1x3 + 100x2x3
Find the set of values of a for which the symmetric bilinear form associated to g' defines a real inner product on
R.

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