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Mathematics 108A Homework # 1 Due: October 1, 2009 1. Show that Z2 = {0,1} is a field. Is Z3 = {0,1,2} a field? Z4 = {0,1,2,3}? 2 Let Q be the rational numbers and Q[√3] = {p + q √3 | p, q Q}. Show that Q[√3] is a vector space over Q. 3. Let X denote the collection fifth degree polynomials p(x) (in variable x with real coefficients) with roots at x = , i.e. p() = 0. Is X a subspace of the vector space of fifth degree polynomials with real coefficients. If yes, explain why. If no, explain why not? 4. Let V be the first quadrant in the xy-plane: that is, let x | V = | x 0, y 0 y | (i) If u and v are in V, is u + v in V? Why or why not? (ii) If u is in V and c is in R, is cu in V? Why or why not? (iii) Is V a real vector space? Explain. 5. Suppose that F is a field. (i) Prove that –(-r) = r, for any rF. (ii) Prove that (-1)(-1)=1. (ii) Prove that if r0, (r--1)-1=r, where r—1is the multiplicative inverse of r. 6. Suppose that W is a vector space and wW. Prove that –(-w) = w. 7. Give an example of a nonempty subset U of R2 that is closed under scalar multiplication but is not a subspace of R2. 8. Prove that the intersection of any collection of subspaces of a vector space is a vector space. 9. Prove that the union to two subspaces of a vector space is a subspace if and only if one of them is contained in the other. 10. Prove or give a counterexample: If U1, U2, and W are subspaces of a vector space V such that U1+W = U2+W, then U1 = U2 . 11. Prove or give a counterexample: If U1, U2, and W are subspaces of a vector space V such that U1W = U2W, then U1 = U2 .