Question

The form x2 + y2 + z2 and x2 - y2 - z2 are equivalent. True or false

Answer 1

It is true because there exists a nonsingular linear transformation

$$X = BY,$$

where X is a matrix of the first quadratic form x²+y²+z² ,

and

$$X=\begin{pmatrix} 1 & 0 & 0\\ 0& 1 & 0\\ 0& 0 & 1 \end{pmatrix}$$

Y is a matrix of the second quadratic form x²-y²-z²

and

$$Y=\begin{pmatrix} 1 & 0 & 0\\ 0& -1 & 0\\ 0& 0 &- 1 \end{pmatrix}$$

and B is a matrix of the nonsingular linear transformation and

$$B=\begin{pmatrix} 1 & 0 & 0\\ 0& -1 & 0\\ 0& 0 &- 1 \end{pmatrix}$$

and

$$Det(B)=1\neq0$$