Problem 1. Show that, if v1, v2, 13 are perpendicular in R" , then (Hint: first show that the analogous formula is true for two perpendicular vectors.) Problem 2. Let v1 = (2, 4, 1, 2), v2 = (1, 1, 1,3), and va = (2, 1, 2, 1) in R4. 1. Find a vector uz in the span of v and v2 that's perpendicular to v1. 2. Find a vector us in the span of v1, v2 and v3 that's perpendicular to v, and v2. Explain why v1, u2, us are mutually perpendicular vectors with the same span as v1, v2, v3. Find the closest vector in the span of v1, v2, 13 to (2, 2, 1, 3). Problem 3. Let T be the linear mapping R? - R? given by the matrix a b A = b (A is called a symmetric matrix). Find the minimal polynomial of 7. Show that T' always has two real (not necessarily distinct) eigenvalues. Show that there is always a basis of eigenvectors for T