1. [5] Suppose V is a vector space with dim(V) = 14, and U and W are vector subspaces of V with dim(U) = 8, dim(W) = 10. (a) What are the possible dimensions of U + W? Explain. (b) What are the possible dimensions of U {'1 W? Explain. (This should remind you of exercise 4.115 on page 161 of your text.) 2. [10] Assuming only that V is a vector space with basis B = {111,v2,123, ...,'vn}, prove that every vector 'u E V has a unique expression as v=clv1+6202+03v3+m+cnvm for (:1, 02, (:3, ..., an E R. (Hints: Use the spanning property of B to (a) establish existence, and use linear independence of B to (b) establish uniqueness. This is a standard exercise. The proof might be in your text, but your instructor could not nd it.) 3. [5] Work one of the following (a) Compute the rref, rank, and nullity of the matrix 5 20 -1 -37 17 0 113 3 1 4 2 52 21 -1 25 -9 M = 0 0 -2 -54 -16 4 4 18 2 8 2 50 26 3 56 3 4 0 2 -5 1 -21 3 You should use a computer to work this. Some assistance is offered in the .tex source file for this document. Don't forget to explain what system you used and give some hints about the code that you used. (b) Determine the coordinate vector of v = (4, 8, 0, 12) with respect to the basis B = {(1, 1, 1, 1), (1, 1, -1, -1), (1, -1, 1, -1), (1, -1, -1, 1) }