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Math 20F Test 1 100 points January 28, 2009 --------------------------------------------------------------------------------------------------Notation: All scalars and matrix entries are in the set R of real numbers. Points: (1),(2) 12 pts each (3A),(3B),(3C),(4) 15 pts each (3D) 10 pts (3E) 6 pts Directions: Show all work and justify all answers, except in problems (1) and (2). ----------------------------------------------------------------------------------------------------(1) Let S = { v1 , . . . , vn } be a set of n vectors in Rm . Complete the following sentences in such a way as to provide precise, explicit definitions: (A) The set S is called linearly dependent if ___________________________. (B) The span of S is defined to be the set ______________________________. -----------------------------------------------------------------------------------------------(2) Let T : Rn Rm be a function (i.e., a transformation). Complete the following sentences in such a way as to provide precise, explicit definitions: (A) T is called onto if _________________________________________________. (B) T is called one-to-one if ___________________________________________. -------------------------------------------------------------------------------------------------(3) Define A 0 0 0 1 A= 1 0 1 0 2 1 5 1 and U and b as follows: 2 1 0 1 0 2 2 , U= 0 1 3 0 1 7 0 0 0 1 2 , -3 b = 1 5 . (Note that the matrices A and U are 3 by 5, while the vector b is 3 by 1.) (A) Show (step by step) that U is the reduced row echelon form of A. (B) Find two 5 by 1 vectors c and d such that Span{c, d} is the set of all solutions x in R5 to the equation Ax = 0 . (C) Find the set of all solutions x in R5 to the equation Ax = b . Hint: Part (B) may be of use. (D) Is every 3 by 1 vector equal to some linear combination of the five columns of A ? Explain your answer. (E) What is the number in the lower right position of the 5 by 5 matrix (AT U)T ? --------------------------------------------------------------------------------------------------(4) Let T : R3 R5 be a one- to-one linear transformation and let L denote a straight line in R3 . Show that the image T(L) in R5 is also a line.