Answer to Question #146822 in Linear Algebra for Sourav Mondal

Question

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

Answer 1

Let's the quadratic form 

$x^2+y^2+z^2$

and

$xz+yz$

let's replace

$x=x_1-z_1\\ y=y_1\\ z=x_1+z_1$

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

let's replace

$x_2=x_1+\frac{1}{2}y_1\\ y_2=\frac{1}{2}y_1-z_1\\ z_2=z_1$

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

True