Suppose that {v1, v2,..., vn} is an orthonormal set in Rn with the standard inner product. Let the matrix A be given by A = v1 v2 ··· vn . Show that A is non- singular and compute its inverse. Give three different ex- amples of such a matrix in R2 or R3.

Let {v1, v2, . .. , vn} be an orthogonal set in R" with respect to the Euclidean inner product. Let A be the n x n matrix with row vectors v1, v2, ... , Vn. b) What extra condition(s) do we need from the vectors v1, v2, ..., Vn so that the matrix AAT is an identity matrix? Justify your answer!

5. Prove: Let A and B be nxn matrices such that AB is singular. Prove that either A or B is singular.

Let S= (V1,., Vk be a set of k vectors in R", with k