Orthogonal Transformations
Orthogonal Matrices
In Exercises 1-2, which of the matrices are orthogonal?
[0.8 0.6]
[0.6 0.8]
[2]
[5 2]
Consider the n x m matrix A, vector v in R^n, and a vector w in R^m. Show that (Av) · w = 6. (A^T w)
Find all orthogonal 2 x 2 matrices. Is there an orthogonal transformation T from R^3 to R^3 such that det(T) = -1?
Find a basis of the space V of all symmetric 3 x 3 matrices and thus determine the dimension of V.
Is the transformation L(A) = A^T from R^(2x2) to R^(2x2) linear? Is L an isomorphism? Find the matrix of the linear transformation L(A) = A^T from R^(2x2) to R^(2x2) with respect to the basis {[1 0], [0 1]}.