SOLUTION: This is my second and last question. This is not in a textbook. a.)Let T be a linear operator on V. SHow that if (Tu|v)=0 for every u,v in V, then T is the zero operator on V.

Algebra ->  College  -> Linear Algebra -> SOLUTION: This is my second and last question. This is not in a textbook. a.)Let T be a linear operator on V. SHow that if (Tu|v)=0 for every u,v in V, then T is the zero operator on V.       Log On


   

Question 36646: This is my second and last question. This is not in a textbook.
a.)Let T be a linear operator on V. SHow that if (Tu|v)=0 for every u,v in V, then T is the zero operator on V.
b.)Prove the Generalized Pythagorean Theorem: Suppose {v1,v2,...,vn} is an orthogonal set of vectors. Then ||v1+v2+...+vn||^2=||v1||^2+...||v2||^2+...+||vn||^2
For (a.) I think you fix u and get it, but I don't understand how.
For (b.) Do I only have to do this:
||v1+v2+...+vn||^2=(v2+v2+...vn|v1+v2+...+vn)
=(v1|v1)+(v2|v2)+...(vn|vn) since the vectors are orthog.
=||v1||^2+...||v2||^2+...+||vn||^2
Is this all i have to do or is there more steps in there?
Answer by venugopalramana(3286) About Me  (Show Source):
You can put this solution on YOUR website!
YOU ARE INTERESTED IN SECOND BIT
YOU BETTER WRITE CLEARLY THAT THE VECTORS BEING ORTOGONAL
V1*VJ=0...IF I IS NOT EQUAL TO J AND
VI*VJ=|VI|^2...IF I=J.....WHICH PROVES THE IDENTITY