Back to Application: Simplification of quadratic forms

Exercise 11.9.1

Avg difficulty 3.0

Which of the following are quadratic forms?

$f_{1}(x,y,z)=(x+y+z)^{2}$ .
$f_{2}(x,y,z)=x^{2}+y^{2}+z^{2}$ .
$f_{3}(x,y,z)=(x+y)^{2}+(x+z)^{2}+(y+z)^{2}$ .
$f_{4}(x,y,z)=x^{2}-y^{2}$ .
$f_{5}(x,y,z)=x^{2}+2xyz+(y+z)^{2}$ .
$f_{6}(x,y,z)=x(y+z)$ .
$f_{7}(x,y,z)=x^{2}y^{2}z^{2}$ .

Selinger, Matrix Theory and Linear Algebra, Inner product spaces, Application: Simplification of quadratic forms, Exercise 11.9.1

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.