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

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- March 17th 2013, 04:51 PMjackGeeSymmetric 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 - March 17th 2013, 07:16 PMchiroRe: Symmetric mappings help!!
Hey jackGee.

Just to be clear, can you give a definition of symmetric mapping in mathematical form? - March 17th 2013, 08:03 PMjackGeeRe: 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 - March 17th 2013, 08:10 PMchiroRe: 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)?

- March 17th 2013, 08:29 PMjackGeeRe: Symmetric mappings help!!
so I have to show <Ax,y> = <x ,Ay> where x and y are eigenvectors?

- March 17th 2013, 08:40 PMjackGeeRe: Symmetric mappings help!!
.

- March 18th 2013, 04:21 AMchiroRe: 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