Consider the basis b = {(1,1,0),(1,0,-1),(2,1,0)} for R^3.
which of the following matrices A =[T]_{bb }define symmetric mappings of R^3?
1 1 0
1 1 0
0 0 1
2 0 0
0 2 0
0 0 2
-1 1 2
1 4 0
2 0 1
Let T : V -----> V be any linear mapping and a={v1,.....,vn} be any orthonormal basis of V
then the matrix A is symmetric iff <T(x),y> = < x ,T(y)> for x,y belong in V
I'm confused by the given basis b because usually I would find a set of eigenvectors and then use it to find an orthonormal basis
then show A is diagonalizable
Personally I'm wondering if you are dealing with an inner product with the following properties:
http://linear.axler.net/Chapter7.pdf