Answer to Question #234681 in Linear Algebra for Anuj

Question #234681

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



1
Expert's answer
2021-09-13T05:41:18-0400

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.


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