Answer to Question #146822 in Linear Algebra for Sourav Mondal

Question #146822

There is no co-ordinate transformation that transforms the quadratic form x² + y²+ z² to xz + yz. True or false


1
Expert's answer
2020-11-30T10:35:02-0500

Let's the quadratic form 

"x^2+y^2+z^2"

and

"xz+yz"

let's replace

"x=x_1-z_1\\\\\ny=y_1\\\\\nz=x_1+z_1"

"xz+yz=(x_1-z_1)(x_1+z_1)+y_1(x_1+z_1)=\\\\\n=x_1^2-z_1^2+x_1y_1+y_1z_1=\\\\\n=(x_1^2+x_1y_1+\\frac{1}{4}y_1^2)-\\frac{1}{4}y_1^2+y_1z_1-z_1^2=\\\\\n=(x_1+\\frac{1}{2}y_1)^2-(\\frac{1}{4}y_1^2-y_1z_1+z_1^2)=\\\\\n=(x_1+\\frac{1}{2}y_1)^2-(\\frac{1}{2}y_1-z_1)^2"

let's replace

"x_2=x_1+\\frac{1}{2}y_1\\\\\ny_2=\\frac{1}{2}y_1-z_1\\\\\nz_2=z_1"

"xz+yz=x^2_2-y^2_2\\neq x^2+y^2+z^2"

True


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