Question

a. Find the orthogonal and normal canonical forms of 2y^2-2yz+2zx-2xy.

b. The operation,* defined by a*b= sin(ab), is a binary operation on N

True or false with full explanation

Answer 1

1.$f(x;y;z)=2y^2-2yz+2zx-2xy=(y^2-2yz+z^2)+(y^2-2xy+x^2)-z^2+2xz-x^2=(y-z)^2+ (y-x)^2-(z-x)^2.$

Let's make the transformation of variables: $y-z=a; y-x=b; z-x=c.$ We have $f(a;b;c)=a^2+b^2-c^2.$ It is the orthogonal and normal canonical form.

2.For all $a,b\in N$ it is exists the unique value $\sin (ab)$ .The domain of the function $y=\sin(x)$ is $R$ , and by the properties of this function $y=\sin(x)$ it is unambiguous. So $"*"$ is a binary operation.