Fixing V and Finding U Now, assume we have 2 users, 3 movies, and a 2 by 3 matrix Y given by First element of U is: Y The second element of U is: = Our goal is to find U and V such that X = UVT closely approximates the observed ratings in Y. Assume we start by fixing V to initial values of [4, 2, 1]. Find the optimal 2 × 1 vector U in this case. (Express your answer in terms of X). 18 ? 2 ? 5

Use a software program or a graphing utility with matrix capabilities to write v as a linear combination of u,, u,, uz, ua, and Then verify your solution. (Enter your answer in terms of u,, u2, U3, U4i and u5.) V = (6, 6, –10, 14, 6) (1, 2, –3, 4, –1) u2 (1, 2, 0, 2, 1) (0, 1, 1, 1, -4) U3 u4 (2, 1, –1, 2, 1) u5 = (0, 2, 2, –1, –1) V =

4

Use a software program or a graphing utility with matrix capabilities to write v as a linear combination of u,, uz, uz, u̟, and ug. Then verify your solution. v = (3, 5, 8, -5, 1) u = (1, 1, -1, 2, 1) u2 = (2, 1, 2, -1, 1) uz = (1, 2, 0, 1, 2) u, (0, 2, 0, 1, -4) (1, 1, 2, -1, 2)