1. Let B = (v1, .... Un), C= {ul,....un}, and D = (w1,..., Win} be three bases for R". (a) Prove the following identity regarding the change of basis matrices: Pc +DPBAC = l'B +D. (b) Consider the matrices Mg and My whose columns are the vectors in B and D, respectively: MB = [v1 ... Un ] and Mp = [ wi ... Win ]. Prove that PEA+D = (MD) ME. 2. Let U be a n x k matrix. Prove that the columns of U are orthonormal if and only if (UP ) U = Ik (the superscript 7' denotes transpose). Assuming that the columns of ( are orthonormal, deduce that: . U "preserves the dot product" in the sense that (Us) . (Uy) = 2 . 7 for all , y ERk. . U "preserves length" in the sense that ||Vall = (| || for all me R*. . If U is square, then / is invertible, U-1 = UT, and | det | =1. (Hint: for any square matrix A, it is true that det A = det A"; do you see why?) . If U is square and V is another n x n matrix with orthonormal columns, then their product UV also has orthonormal columns. (Orthonormal matrices are often used in applications because of these properties!) 3. Use the method of least-squares to find the parabola (d = 2 in the notation from class) that best fits the four data points (-1, 1), (1, -1), (2,3), and (3,6). (You may find yourself wanting to know the inverse of a 3 x 3 matrix; you can use an online calculator for this, but clearly indicate where you use it!) 4. Find the singular value decomposition of A JOHN Make sure to show all your steps