CS 110 Berkeley

Solutions to Homework 1. Math 110, Fall 2006. Prob 1.2.1. (a) True this is axiom (VS 3). (b) False we proved the zero vector is unique. (c) False: take x to be the zero vector, and take any scalars a and b. (d) False: take a to be the zero scalar a

Solutions to Homework 1. Math 110, Fall 2006. Prob 1.2.1. (a) True this is axiom (VS 3). (b) False we proved the zero vector is unique. (c) False: take x to be the zero vector, and take any scalars a and b. (d) False: take a to be the zero scalar a

Solutions to Homework 2. Math 110, Fall 2006. Prob 1.4.4. (a) Yes, since the linear system a+b 2a + 3b -a a-b has a solution a = 3, b = -2. (b) No, the corresponding linear system has no solution. (c) Yes, the corresponding linear system has a soluti

Solutions to Homework 3. Math 110, Fall 2006. Prob 2.1.10. By the linearity of T , we use the fact (1, 0) + 3(1, 1) = (2, 3) to obtain T (2, 3) = T (1, 0) + 3T (1, 1) = (1, 4) + 3(2, 5) = (5, 11). The map T is 1 1 as we see that T (a(1, 0) + b(1, 1)

Solutions to Homework 4. Math 110, Fall 2006. Prob 2.3.13. Let A = [aij ]. Then At = [aji ], and n n tr(A) = i=1 aii = j=1 ajj = tr(At ). The elements of A = [aij ], B = [bij ], AB = [cij ] and BA = [dij ] are connected by the formulas n n cij =