Problem 1. (1 point) Let 6 8 12 $1- 6 ,V2 20 ,V3 21 2 ~12 7 Is the set {V1, v2, V3} linearly independent? 0 A. Yes 0 B. No If the vectors are linearly dependent, nd coefficients, not all zero, that make the following equation true. If the vectors are linearly independent, enter zero in every answer blank, since those are only the values that make the equation true. 0 6 8 12 0 = 76 + 720 + 721 0 2 12 7 Problem 2. (1 point) Suppose that V1, . . ., Vn are vectors in Rm. Which of the following sentences are equivalent to the statement " The vectors v1, ..., Vn span R" " ? Select all correct answers. A. Every b E R" is a linear combination of V1, . . . , Vn. O B. Span {v1, . . ., Vn} = Rm O C. For only some (not all) b E R , the equation xivIt . . . + En Vn = b has a solution. D. Every b E R" is a linear combination of V1, . . . , Vn. E. There exists only finitely many b E RR" that is a linear combination of V1, . . . , Vn. O F. For every b E R , the equation x1 Vit . . . + an Vn = b has a solution. O G. For every b E R", the equation xcvi + . . . + an Vn = b has a solution. OH. {CIVI t . . . + Cn Vn : C1, . .., Cn E R} = RmProblem 3. (1 point) Aset of two, three, or four red vectors in R2 or R3 is shown in each picture below. Determine whether each set of vectors is linearly independent or linearly dependent. Note: For the pictures in R3, a grid is drawn in the ray-plane, and vectors with their tip on the grid are in the my-plane, while vectors with their tip not on the grid are not in the ray-plane. z z z I I I i choose v choose V' i choose V' Z y y :17 11 :1: ' t , i choose v choose v i ' choose v Problem 4. (1 point) Given that the reduced row echelon form of 2 N H 9 -2 1 -6 O is 1 0 0 3 0 0 1 0 1 1 9 if possible, write -6 as a linear combination of the vectors 9 2 6 + 2 + 4 OProblem 5. (1 point) Which of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.) iii'i'fi'i} Cl 8 7 9 E. 5 , 2 , 6 0 0 0 C1 3 8 5 F. 4 , 3 , 7 7 5 12 Problem 6. (1 point) The vectors are linearly independent if and only if k 75 3 Problem 7. (1 point) Let u, v, w be three linearly independent vectors in R7. Determine whether the following sets of vectors are linearly independent or dependent. ' Select an Answer V 1. The set {u + 12,1) + 10,112 + u} ' Select an Answer V 2. The set {u 11,1) w, w u} Problem 8. (1 point) Determine which of the following sets of vectors are linearly independent and which are linearly dependent. Select an answer V 1. 5 10 -4 -8 -5 10 Select an answer V 2 . 6 -3 9 -9 8 W -4 -6 -7 -3 -2 -6 -2 -8 5 3 4 Select an answer V 3. -3 3 -3 .9 Select an answer V 4. 7 -8 0 -3 -8 0 5 -10 2 0 -8 7 6 -10\fProblem 9. (1 point) Do the columns of the matrix span R3? Select an Answer 2 -2 -6 2 A = 7 8 23 -11 7 7 21 -7 Select an Answer V 2. 9 -4 A = 2 2 -1 -1 Select an Answer 3. 1 1 -5 A = -3 -2 13 5 5 -20 Select an Answer V 4. 8 16 - A = 2 4 -2 3 6 -3Problem 10. (1 point) Let W be the set of all vectors of the form t II at H 5a + 4b a b . Find vectors 1? and '3 in R3 such that W = span {12, '3 . Problem 11. 4t 3s (1 point) Let W be the set of all vectors of the form 43 5t . Find vectors {1: and 5 in R3 such that W = span {13, i5}. 3.s 4t PM || :21 ||