MATH 3319-001 Sample Exam 1b, Fall 2022 Solve the following problems in the space provided. Show your work /reasoning and indicate your final answer clearly. Indicate problems continued on additional sheets of paper and label those sheets carefully with name and problem number. No calculators, cell phones, or other technology permitted. You may use one 5"x7" card with information written on it. 1. The 3x3 Magic Squares problem (4.6 #29): Give a basis for the vector space of 3x3 matrices for which each row and each column has the same sum. Think about how many entries you can choose arbitrarily before the constraint of every row and column having the same sum then chooses the remaining entries automatically. 0 3 1 2. Give a basis for the nullspace of 0 0 -3 2 0 1 1 0 3. Given fi(x) = 1 + x, f2(x) = 1-x2, (a) give f3(x) E P2(x) such that {f1, f2, f3} forms a basis for P2, and prove it is a basis; (b) give g3(x) E P2(x), 93 / 0, such that {f1, f2, 93} does not form a basis for P2, and prove it is not a basis. 4. Give a linearly dependent set containing: (a) one vector in R3; (b) 2 nonzero vectors in R3; (c) 3 vectors in R3, any 2 of which are linearly independent. 5. (a) For what value(s) of a are the vectors {(1, -1, 2), (-4, 0, 1), (1, 1, x) } coplanar? (b) When they are coplanar, what specific linear combination of the first two vectors gives (1, 1, x)? 6. True or false? Correct any false statements. (a) The determinant of a matrix whose elements are all positive must be positive. (b) There exists a spanning set for R which contains 7 nonzero vectors. [Alternate last question] Give two genuinely different reasons why a matrix might not have an inverse, and an example of each