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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 27EQ
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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.
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.
(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.
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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