Matrix Representation : Orthogonal Transformation

Please see the attached file for full problem description. 1. Show that the 3 x 3 matrix P= [ 1/2 -1//2 0 ] [ 0 0 1 ] [ 1/2 1/2 0 ] is an orthogonal matrix. Show work. Help: : is the square root of

Matrix Representation : Orthogonal Transformation and Determinant

Please see the attached file for full problem description. Write a proof for the following statement: If P is an n x n orthogonal matrix, then det(P) = 1 or –1. Show work. Help: det: is the determinant

Eigenvalues of a Transformation

The linear operator T: R^3 -> R^3 defined by T(x_1, x_2, x_3) = (x_1 - 3x_3, x_1 + 2x_2 + x_3, x_3 - 3x_1). Find the eigenvalues of the transformation T. Show work. (See attachment for the full question.)

Eigenspaces; transformations

Please see the attached file for full problem description.

Transformations : Diagonalization of Matrices

Please see the attached file for full problem description. The linear operator T: R^3 R^3 defined by T(x_1, x_2, x_3) = (x_1 - 3x_3, x_1 + 2x_2 + x_3, x_3 – 3x_1). Determine whether or not there is a basis F for R^3 relative to which the transformation T can be represented by a diagonal matrix D=[T]_F. If there is, show that D is similar to the standard matrix representation [T]_E. If not, why? Show work. Help: R^3: is Euclidean 3-space