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MATH 304 Linear Algebra Practice Exam

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Math 304-6 Feingold Linear Algebra Spring 2020 Final Exam NAME (Printed): P roblem : 1 2 3 4 5 6 T otal V alue : 20 30 30 20 20 30 150 Score : SHOW ALL NECESSARY WORK FOR EACH PROBLEM. m m Notations: Rm is the zero matrix. The n is the set of all m ⇥ n real matrices and 0 2 R T transpose of matrix A is denoted by A . n m (1) (20 Points) For A 2 Rm be the linear function LA (X) = AX. n , let LA : R ! R Suppose A row reduces to C in RREF and Rank(A) = r. Fill in the blanks to answer the following questions in terms of m, n and r if needed. No justifications are required. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) The value of dim(Ker(LA )) = The value of dim(Range(LA )) = LA is injective precisely when LA is surjective precisely when The relation between m and n guarantees that LA is not surjective. The relation between m and n guarantees that LA is not injective. If LA is surjective, you can be sure that the relation between m and n is If LA is injective, you can be sure that the relation between m and n is If m = n and LA is invertible, you can be sure that C = If m = n and C has a zero row, you can be sure that LA is (2) (30 Points) Let L : R4 ! R22 be the linear map (a + b + c) (a + 2b + 3d) L([a b c d]) = . (b + c d) (a + 2b + c + d) (a) (b) (c) (d) (10 pts) Find the set of all vectors in Ker(L), a basis for Ker(L), and dim(Ker(L)). (8 pts) Find all vectors in Range(L), a basis for Range(L), and dim(Range(L)). (6 pts) Is L one-to-one? Explain why. Is L onto? Exp ...
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