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**1) Let f: V x V -> F be a symmetric bilinear form on V, where F is a field.**

Suppose B={v1,...,vn} is an orthogonal basis for V

This implies f(vi,vj)=0 for all i not=j

=>A=diag{a1,...,an} and we say that f is diagonalized.

Suppose B={v1,...,vn} is an orthogonal basis for V

This implies f(vi,vj)=0 for all i not=j

=>A=diag{a1,...,an} and we say that f is diagonalized.

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Now I don't understand the red part, i.e. how does orthongality imply f(vi,vj)=0 for all i not=j?

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**2) For the following symmetric matrix A,**

A=[1 -1 -1

-1 1 -1

-1 -1 1]

a) find an orthogonal matrix P such that (P^T)AP is diagonal.

b) Let f be the bilinear form on R^3 that has matrix A relative to the basis B={(1,1,1),(1,0,1),(0,1,-1)}. Use the matrix P from part a to find a basis of R^3 relative to which f is represented by a diagonal matrix. Also write out the corresponding diagonalized expression for f.

A=[1 -1 -1

-1 1 -1

-1 -1 1]

a) find an orthogonal matrix P such that (P^T)AP is diagonal.

b) Let f be the bilinear form on R^3 that has matrix A relative to the basis B={(1,1,1),(1,0,1),(0,1,-1)}. Use the matrix P from part a to find a basis of R^3 relative to which f is represented by a diagonal matrix. Also write out the corresponding diagonalized expression for f.

I got part a),

P= [1/sqrt3 1/sqrt2 1/sqrt6

1/sqrt3 -1/sqrt2 1/sqrt6

1/sqrt3 0 -2/sqrt6]

is orthogonal such that (P^T)AP=diag{-1,2,2} is diagonal.

Now part b is the headache, I don't even know how to start, can someone please help me? It seems really challenging, but I am sure someone here knows how to solve it.

Thanks a million!