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Unformatted text preview: Section 3.1 2. Suppose the multiplication c x is defined to produce ( cx 1 , 0) instead of ( cx 1 ,cx 2 ), which of the eight conditions are not satisfied? Solution: 3, 4, 5 10. Which of the following subsets of R 3 are actually subspaces? (a) The plane of vectors ( b 1 ,b 2 ,b 3 ) with b 1 = b 2 . (b) The plane of vectors with b 1 = 1. (c) The vectors with b 1 b 2 b 3 = 0. (d) All linear combinations of v = (1 , 4 , 0) and w = (2 , 2 , 2). (e) All vectors that satisfy b 1 + b 2 + b 3 = 0. (f) All vectors with b 1 ≤ b 2 ≤ b 3 . Solution: a, d, e 19. Describe the column spaces (lines or planes) of these particular matrices: A = 1 2 0 0 0 0 and B = 1 0 0 2 0 0 and C = 1 0 2 0 0 0 . Solution: C ( A ) is a line (the xaxis), C ( B ) is a plane (the xyplane), C ( C ) is a line (through and (1 , 2 , 0)). Section 3.2 1. Reduce these matrices to their ordinary echelon forms U : (a) A = 1 2 2 4 6 1 2 3 6 9 0 0 1 2 3 (b) B = 2 4 2 0 4 4 0 8 8 . Solution: (a) U A = 1 2 2 4 6 0 0 1 2 3 0 0 0 0 0 (b) U B = 2 4 2 0 4 4 0 0 0 . 3. By combining the special solutions in Problem 2, describe every solution to A x = and B x = . The nullspace contains only x = when there are no . Solution: All solutions to A x = are given by x = x 2 ( 2 , 1 , , , 0) + x 4 (0 , , 2 , 1 , 0) + x 5 (0 , , 3 , , 1), all solutions to B x = are given by x = x 3 (1 , 1 , 1). The nullspace contains only x = when there are no free variables . 9. True or false (with reason if true or example to show it is false): (a) A square matrix has no free variables. (b) An invertible matrix has no free variables. (c) An m by n matrix has no more than n pivot variables. (d) An m by n matrix has no more than m pivot variables. 1 2 Solution: (a) False, e.g., A = 0 0 0 0 has two free variables. (b) True, because an invertible matrix has only in its nullspace. (c) True because there are only n variables total. (d) True because every row contains at most one pivot. 31. If the nullspace of A consists of all multiples of x = (2 , 1 , , 1), how many pivots appear in U ? What is R ? Solution: 3 pivots appear in U , and R = I F in block form, where I is the 3 by 3 identity matrix, and F =  2 1 . The 0 blocks have 3 and 1 columns, respectively, and they could have any number of rows. Section 3.3 2. Find the reduced row echelon form R and the rank of these matrices: (a) The 3 by 4 matrix with all entries equal to 4. (b) The 3 by 4 matrix with a ij = i + j 1. (c) The 3 by 4 matrix with a ij = ( 1) j . Solution: (a) Rank 1, R = 1 1 1 1 0 0 0 0 0 0 0 0 . (b) Rank 2, R = 1 0 1 2 0 1 2 3 0 0 ....
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 Spring '08
 Neely
 Linear Algebra

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