# Defining symmetric mappings of R^3

1. Nov 17, 2012

### trap101

Consider the basis β = {(1,1,0), (1,0,-1) , (2,1,0)} for R3. Which of the following matrices A = [T]ββ (where T is the transformation) define symmetric mappings of R3?

Attempt/ Issue: The properties that I know that define a matrix being symmetric are that <T(X), Y> = <X,T(Y)> i.e the innner products of the vectors x,y where x,y are in the vector space.

What I'm tempted to do is multiply each of the basis vectors above by the matrix given and take the inner product with each of the the vectors not being multiplied by the matrix and compare them. So for example take (1,1,0) multiply it by the matrix, then take the inner product of my result with (1,0,-1). Then do the same procedure but instead multiplying (1,0,-1) by the matrix and then taking the inner product with (1,1,0). Compare them and proceed with the other combinations of basis vectors given.

Is that what they're asking of me? That's sort of where I'm having the issue. Because I don't know of many other ways of showing if the matrices are symmetric while using that basis. Of course I could diagonalize the matrices but I don't think that's what I'm meant to do.

Thanks.