Defining symmetric mappings of R^3

[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.

 

[mighty2000]

Dear poster,

Thank you for your question. Your approach of using the inner product to determine if the matrices are symmetric is a valid one. However, there are a few other ways you can show if a matrix is symmetric using the given basis.

1. Use the definition of a symmetric matrix: A matrix A is symmetric if it is equal to its transpose, i.e. A = A^T. So for each of the given matrices, you can calculate its transpose and see if it is equal to the original matrix.

2. Check if the matrix is diagonalizable: A matrix is symmetric if and only if it is diagonalizable. So you can calculate the eigenvalues and eigenvectors of each matrix and see if it is diagonalizable.

3. Use the basis transformation property: The transformation matrix [T]ββ is symmetric if and only if the basis vectors are eigenvectors of the matrix A. So you can calculate the eigenvalues and eigenvectors of each matrix and see if they match with the basis vectors given.

I hope this helps. Let me know if you have any further questions. Good luck!