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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F95d4b0a8-938f-4cc4-9205-fc175a148a5e%2F4f2b67cb-ba96-4ad2-8edf-fe7dce02b961%2Fyg5fkwu_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
Transcribed Image Text:(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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