me me ft-uiowa-math2550 Assignment HW6fall14 due 10/09/2014 at 11:59pm CDT The columns of A could be either linearly dependent or lin- 1. (1 pt) Library/WHFreeman/Holt linear algebra/Chaps 1-4- early independent depending on the case. /2.3.40.pg Correct Answers: • B n Let S be a set of m vectors in R with m > n. Select the best statement. 3. (1 pt) Library/WHFreeman/Holt linear algebra/Chaps 1-4- /2.3.42.pg • A. The set S is linearly independent, as long as it does Let A be a matrix with more columns than rows. not include the zero vector. Select the best statement. • B. The set S is linearly dependent. • C. The set S is linearly independent, as long as no vec- • A. The columns of A are linearly independent, as long tor in S is a scalar multiple of another vector in the set. as they does not include the zero vector. • D. The set S is linearly independent. • B. The columns of A could be either linearly dependent • E. The set S could be either linearly dependent or lin- or linearly independent depending on the case. early independent, depending on the case. • C. The columns of A must be linearly dependent. • F. none of the above • D. The columns of A are linearly independent, as long Solution: (Instructor solution preview: show the student so- as no column is a scalar multiple of another column in lution after due date. ) A • E. none of the above SOLUTION Solution: (Instructor solution preview: show the student so- By theorem 2.13, a linearly independent set in n can contain R lution after due date. ) no more than n vectors. Correct Answers: SOLUTION • B Since there are more columns than rows, when we row re- 2. (1 pt) Library/WHFreeman/Holt linear algebra/Chaps 1-4- duce the matrix not all columns can have a leading 1. /2.3.41.pg The columns of A must be linearly dependent. Correct Answers: Let A be a matrix with more rows than columns. • C Select the best statement. 4. (1 pt) local/Library/UI/MatrixAlgebra/Euclidean/2.3.43.pg • A. The columns of A are linearly independent, as long Let A be a matrix with linearly independent columns. as no column is a scalar multiple of another column in Select the best statement. A • B. The columns of A could be either linearly dependent • A. The equation Ax = 0 always has nontrivial solutions. or linearly independent depending on the case. • B. There is insufficient information to determine if such • C. The columns of A are linearly independent, as long an equation has nontrivial solutions. as they does not include the zero vector. • C. The equation Ax = 0 has nontrivial solutions pre- • D. The columns of A must be linearly dependent. cisely when it has more rows than columns. • E. The columns of A must be linearly independent. • D. The equation Ax = 0 never has nontrivial solutions. • F. none of the above • E. The equation Ax = 0 has nontrivial solutions pre- Solution: (Instructor solution preview: show the student so- cisely when it has more columns than rows. lution after due date. ) • F. The equation Ax = 0 has nontrivial solutions pre- cisely when it is a square matrix. SOLUTION • G. none of the above The zero matrix is an example where the columns are linearly Solution: (Instructor solution preview: show the student so- dependent. The matrix where the top square portion is the iden- lution after due date. ) tity matrix and the portion below that is all zeros is an example where the columns are linearly independent. SOLUTION 1 The linear independence of the columns does not change • D. fu1;u2;u3;u4g is a linearly independent set of vec- with row reduction. Since the columns are linearly independent, tors unless u4 is a linear combination of other vectors after row reduction, each column contains a leading 1. We get in the set. nontrivial solutions when we have columns without a leading 1 • E. fu1;u2;u3;u4g is always a linearly dependent set of in the row reduced matrix. vectors. The equation Ax = 0 never has nontrivial solutions. • F. none of the above Correct Answers: Solution: (Instructor solution preview: show the student so- • D lution after due date. ) 5. (1 pt) local/Library/UI/MatrixAlgebra/Euclidean/2.3.44.pg SOLUTION If the zero vector is a nontrivial linear combination of a vec- Let A be a matrix with linearly independent columns. tors in a smaller set, then it is also a nontrivial combination of Select the best statement. vectors in a bigger set containing those vectors. fu1;u2;u3;u4g is always a linearly dependent set of vectors. • A. The equation Ax = b has a solution for all b precisely Correct Answers: when it has more columns than rows. • E • B. The equation Ax = b always has a solution for all b. 7. (1 pt) Library/WHFreeman/Holt linear algebra/Chaps 1-4- • C. The equation Ax = b has a solution for all b precisely /2.3.47.pg when it is a square matrix. • D. The equation Ax = b never has a solution for all b. Let fu1;u2;u3;u4g be a linearly independent set of vectors. • E. There is insufficient information to determine if Ax = Select the best statement. b has a solution for all b. • F. The equation Ax = b has a solution for all b precisely • A. fu1;u2;u3g is never a linearly independent set of when it has more rows than columns. vectors. • G. none of the above • B. fu1;u2;u3g could be a linearly independent or lin- Solution: (Instructor solution preview: show the student so- early dependent set of vectors depending on the vectors lution after due date. ) chosen. • C. fu1;u2;u3g is always a linearly independent set of SOLUTION vectors. A linear equation has a solution when the row reduced form • D. none of the above of the augmented solution does not have a leading 1 in the ex- Solution: (Instructor solution preview: show the student so- tra column that corresponds to constants. Since the columns of lution after due date. ) the matrix are linearly independent the number or columns is no more than the number of rows. If there are fewer columns than SOLUTION rows we can produce a b for which there is no solution. If the zero vector cannot be written as a nontrivial linear com- The equation Ax = b has a solution for all b precisely when bination of a vectors in a smaller set, then it is also not a nontriv- it is a square matrix. ial combination of vectors in a proper subset of those vectors. Correct Answers: fu1;u2;u3g is always a linearly independent set of vectors. • C Correct Answers: • C 6. (1 pt) Library/WHFreeman/Holt linear algebra/Chaps 1-4- 8. (1 pt) Library/WHFreeman/Holt linear algebra/Chaps 1-4- /2.3.46.pg /2.3.49.pg Let fu1;u2;u3g be a linearly dependent set of vectors. Let u4 be a linear combination of fu1;u2;u3g. Select the best statement. Select the best statement. • A. fu1;u2;u3;u4g is a linearly independent set of vec- • A. fu1;u2;u3;u4g could be a linearly dependent or lin- tors unless u4 = 0. early dependent set of vectors depending on the vectors • B. fu1;u2;u3;u4g could be a linearly independent or chosen. linearly dependent set of vectors depending on the vec- • B. fu1;u2;u3g is a linearly dependent set of vectors un- tors chosen. less one of fu1;u2;u3g is the zero vector. • C. fu1;u2;u3;u4g is always a linearly independent set • C. fu1;u2;u3;u4g is always a linearly independent set of vectors. of vectors. 2 • D. fu ;u ;u ;u g is never a linearly independent set of 1 2 3 4 2 3 vectors. T • E. fu1;u2;u3g is never a linearly dependent set of vec- A = 4 5. tors. Correct Answers: • F. fu1;u2;u3g could be a linearly dependent or linearly dependent set of vectors depending on the vector space • 2 3 chosen. −10 5 −7 • G. none of the above 4 −3 −2 −8 5 3 1 −10 Solution: (Instructor solution preview: show the student so- lution after due date. ) • 2 1 0 3 3 SOLUTION 4 −2 −4 −1 5 If u4 = a1u1 + a2u2 + a3u3, then 1 −1 −2 0 = a1u1 + a2u2 + a3u3 − u4 ”fu1;u2;u3;u4g is never a linearly independent set of vec- tors.” 13. (1 pt) Library/Rochester/setAlgebra34Matrices/cubing 2x2.pg 4 -3 Correct Answers: Given the matrix A = , find A3. 0 3 • D A3 = 9. (1 pt) Library/Rochester/setLinearAlgebra9Dependence- . /ur la 9 7.pg Correct Answers: • 64 The vectors 2 3 2 3 2 3 • -111 -1 -4 -3 • 0 v =4 0 5 , u =4 -12 5 , and w =4 -7 5 . • 27 3 31 +k 16 are linearly independent if and only if k 6= . 14. (1 pt) Library/maCalcDB/setLinearAlgebra3Matrices/ur la 3 6.pg Correct Answers: If A and B are 6 × 2 matrices, and C is a 4 × 6 matrix, which of • -7 the following are defined? 10. (1 pt) Library/TCNJ/TCNJ LinearIndependence/problem3.pg • A. C + B • B. CA If k is a real number, then the vectors (1;k);(k;3k + 40) are • C. B + A linearly independent precisely when • D. BT k 6= a;b, • E. BTCT where a = , b = , and a < b.