# Symmetric mappings help!!

• Mar 17th 2013, 03:51 PM
jackGee
Symmetric mappings help!!
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
• Mar 17th 2013, 06:16 PM
chiro
Re: Symmetric mappings help!!
Hey jackGee.

Just to be clear, can you give a definition of symmetric mapping in mathematical form?
• Mar 17th 2013, 07:03 PM
jackGee
Re: Symmetric mappings help!!
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
• Mar 17th 2013, 07:10 PM
chiro
Re: Symmetric mappings help!!
For these examples, can you just show whether the condition holds by evaluating each example? (In other words, evaluate T(x), T(y) and the inner products given an x, y from the basis corresponding to V)?
• Mar 17th 2013, 07:29 PM
jackGee
Re: Symmetric mappings help!!
so I have to show <Ax,y> = <x ,Ay> where x and y are eigenvectors?
• Mar 17th 2013, 07:40 PM
jackGee
Re: Symmetric mappings help!!
.
• Mar 18th 2013, 03:21 AM
chiro
Re: Symmetric mappings help!!
Personally I'm wondering if you are dealing with an inner product with the following properties:

http://linear.axler.net/Chapter7.pdf