ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

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1 me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following systems of equations x + y =?. 2 x + 2 y = 2 28 x + 28 y = 28 5x + y = 7? 2. 2x 5y = 2 7x + 23y = 3 5x + y = 7? 3. 2x 5y = 2 7x + 23y =. ( pt) UI/Fall/lin span.pg Let A = 3-8, B = 3 -, and C = Which of the following best describes the span of the above 3 vectors?. -dimensional point in R 3. -dimensional line in R 3 C. 2-dimensional plane in R 3 D. R 3 Three identical lines Determine whether or not the three vectors listed above are Three lines intersecting at a single point linearly independent or linearly dependent. Three non-parallel lines with no common intersection. 2. ( pt) Library/TCNJ/TCNJ MatrixEquations/problem.pg Let A = and x = ?. What does Ax mean? Linear combination of the columns of A 3. ( pt) Library/WHFreeman/Holt linear algebra/chaps -- / pg Assume {u,u 2,u 3 } spans R 3. Select the best statement.. {u,u 2,u 3,u } spans R 3 unless u is a scalar multiple of another vector in the set.. {u,u 2,u 3,u } never spans R 3. C. {u,u 2,u 3,u } spans R 3 unless u is the zero vector. D. There is no easy way to determine if {u,u 2,u 3,u } spans R 3. E. {u,u 2,u 3,u } always spans R 3. F. none of the above The span of {u,u 2,u 3 } is a subset of the span of {u,u 2,u 3,u }, so {u,u 2,u 3,u } always spans R 3. E. linearly dependent. linearly independent If they are linearly dependent, determine a non-trivial linear relation. Otherwise, if the vectors are linearly independent, enter s for the coefficients, since that relationship always holds. A+ B+ C =. C 2; -; 5. ( pt) local/library/tcnj/tcnj BasesLinearlyIndependentSet- /3.pg Check the true statements below:. The columns of an invertible n n matrix form a basis for R n.. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for ColA. C. The column space of a matrix A is the set of solutions of Ax = b. D. If H = Span{b,...,b p }, then {b,...,b p } is a basis for H. E. A basis is a spanning set that is as large as possible.

2 . ( pt) local/library/ui/..23.pg 7 Find the null space for A =. What is null(a)?. span, { +7 } + +. span +7 C. R 2 D. span +7 + E. span + +7 { } +7 F. span + G. R 3 H. none of the above. A is row reduced. The basis of the null space has one element for each column without a leading one in the row reduced matrix. Thus Ax = has a one dimentional null space, and thus, null(a) is the subspace generated by 7. D 7. ( pt) local/library/ui/fall/hw7 2.pg Suppose that A is a 8 9 matrix which has a null space of dimension 5. The rank of A=. - Using the Rank-Nullity theorem, if the dimensions of A is n x m, rank(a) = m - nullity(a) = 9-5 = I 8. ( pt) local/library/ui/fall/hw8 5.pg Find the determinant of the matrix -5 A = det(a) = G. H. 2 I. 3 J. K. None of those above G If A is an m n matrix and if the equation Ax = b is inconsistent for some b in R m, then A cannot have a pivot position in every row.. True If the equation Ax = b is inconsistent, then b is not in the set spanned by the columns of A.. True 2

3 . ( pt) local/library/ui/fall/volume.pg Find the volume of the parallelepiped determined by vectors 5 3, 2, and E. ( pt) local/library/ui/problem7.pg A and B are n n matrices C. - D. -2 E. - F. G. I. 5 J. 7 K. None of those above Adding a multiple of one row to another does not affect the determinant of a matrix.. True If the columns of A are linearly dependent, then deta =. Suppose A x = has an infinite number of solutions, then given a vector b of the appropriate dimension, A x = b has. No solution. Unique solution C. Infinitely many solutions D. at most one solution E. either no solution or an infinite number of solutions F. either a unique solution or an infinite number of solutions G. no solution, a unique solution or an infinite number of solutions, depending on the system of equations H. none of the above E Suppose A is a square matrix and A x = has an infinite number of solutions, then given a vector b of the appropriate dimension, A x = b has. No solution. Unique solution C. Infinitely many solutions D. at most one solution E. either no solution or an infinite number of solutions F. either a unique solution or an infinite number of solutions G. no solution, a unique solution or an infinite number of solutions, depending on the system of equations H. none of the above 3. True det(a + B) = deta + detb.. True The vector b is in ColA if and only if A v = b has a solution. True The vector v is in NulA if and only if A v =. True

4 If x and x 2 are solutions to A x =, then 5 x + x 2 is also a solution to A x =.. True Hint: (Instructor hint preview: show the student hint after attempts. The current number of attempts is. ) Is NulA a subspace? Is NulA closed under linear combinations? If x and x 2 are solutions to A x = b, then 3 x + 9 x 2 is also a solution to A x = b.. True Hint: (Instructor hint preview: show the student hint after attempts. The current number of attempts is. ) Is the solution set to A x = b a subspace even when b is not? Is the solution set to A x = b closed under linear combinations even when b is not? Find the area of the parallelogram determined by the vectors and J. 5 I Suppose A is a 5 3 matrix. Then nul A is a subspace of R k where k = H Suppose A is a 7 matrix. Then col A is a subspace of R k where k =. - I 22. ( pt) Library/Rochester/setLinearAlgebraInverseMatrix- /ur la 2.pg 8 The matrix is invertible if and only if k. 9 k ( pt) Library/Rochester/setLinearAlgebra9Dependence- /ur la 9 7.pg Thevectors v = -, u = 2 -, and w = k are linearly independent if and only if k ( pt) Library/Rochester/setLinearAlgebra23QuadraticForms- /ur la 23 2.pg Find the eigenvalues of the matrix M = Enter the two eigenvalues, separated by a comma:. - Classify the quadratic form Q(x) = x T Ax :. Q(x) is indefinite. Q(x) is positive definite

5 C. Q(x) is negative semidefinite D. Q(x) is negative definite E. Q(x) is positive semidefinite -5, 25. ( pt) Library/Rochester/setLinearAlgebra23QuadraticForms- /ur la 23 3.pg Thematrix A = has three distinct eigenvalues, λ < λ 2 < λ 3, λ =, λ 2 =, λ 3 =. Classify the quadratic form Q(x) = x T Ax :. Q(x) is negative definite. Q(x) is positive semidefinite C. Q(x) is negative semidefinite D. Q(x) is positive definite E. Q(x) is indefinite E 2. ( pt) Library/TCNJ/TCNJ BasesLinearlyIndependentSet- /problem5.pg Let W be the set:,,. Determine if W is a basis for R 3 and check the correct answer(s) below.. W is not a basis because it does not span R 3.. W is not a basis because it is linearly dependent. C. W is a basis. Let W 2 be the set:,,. Determine if W 2 is a basis for R 3 and check the correct answer(s) below.. W 2 is not a basis because it does not span R 3.. W 2 is not a basis because it is linearly dependent. C. W 2 is a basis. C B ( pt) Library/WHFreeman/Holt linear algebra/chaps -- /2.3.2.pg Let A be a matrix with more columns than rows. Select the best statement.. The columns of A could be either linearly dependent or linearly independent depending on the case.. The columns of A are linearly independent, as long as they does not include the zero vector. C. The columns of A are linearly independent, as long as no column is a scalar multiple of another column in A D. The columns of A must be linearly dependent. E. none of the above Since there are more columns than rows, when we row reduce the matrix not all columns can have a leading. The columns of A must be linearly dependent. D 28. ( pt) Library/WHFreeman/Holt linear algebra/chaps -- /2.3..pg Let {u,u 2,u 3 } be a linearly dependent set of vectors. Select the best statement.. {u,u 2,u 3,u } is a linearly independent set of vectors unless u is a linear combination of other vectors in the set.. {u,u 2,u 3,u } could be a linearly independent or linearly dependent set of vectors depending on the vectors chosen. C. {u,u 2,u 3,u } is always a linearly dependent set of vectors. D. {u,u 2,u 3,u } is a linearly independent set of vectors unless u =. E. {u,u 2,u 3,u } is always a linearly independent set of vectors. F. none of the above If the zero vector is a nontrivial linear combination of a vectors in a smaller set, then it is also a nontrivial combination of vectors in a bigger set containing those vectors. {u,u 2,u 3,u } is always a linearly dependent set of vectors.

6 C 29. ( pt) Library/WHFreeman/Holt linear algebra/chaps -- /2.3.7.pg Let {u,u 2,u 3,u } be a linearly independent set of vectors. Select the best statement.. {u,u 2,u 3 } is always a linearly independent set of vectors.. {u,u 2,u 3 } could be a linearly independent or linearly dependent set of vectors depending on the vectors chosen. C. {u,u 2,u 3 } is never a linearly independent set of vectors. D. none of the above If the zero vector cannot be written as a nontrivial linear combination of a vectors in a smaller set, then it is also not a nontrivial combination of vectors in a proper subset of those vectors. {u,u 2,u 3 } is always a linearly independent set of vectors. 3. ( pt) Library/WHFreeman/Holt linear algebra/chaps -- /..27.pg Find the null space for A = What is null(a)?.. R 3. R 2 { } 7 C. span D. span 3 E. span { } F. { } 7 G. span { } H. span 7 I. none of the above A is row reduces to. The basis of the null space has one element for each column without a leading one in the row reduced matrix. Thus Ax = has a zero dimentional null space, and null(a) is the zero vector. F 3. ( pt) Library/WHFreeman/Holt linear algebra/chaps -- /.3.7.pg Indicate whether the following statement is true or false.?. If A and B are equivalent matrices, then col(a) = col( B). : FALSE. Consider A = F and B = 32. ( pt) Library/maCalcDB/setLinearAlgebra3Matrices/ur la 3.pg If A and B are 3 2 matrices, and C is a 3 matrix, which of the following are defined?. AC. CA C. C T D. A T C T E. A + B F. C + B CDE 33. ( pt) UI/DIAGtfproblem.pg A, P and D are n n matrices. Check the true statements below:. If A is invertible, then A is diagonalizable.. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. C. If A is diagonalizable, then A has n distinct eigenvalues. D. A is diagonalizable if A has n distinct eigenvectors.

7 E. A is diagonalizable if A has n distinct linearly independent eigenvectors. F. If A is symmetric, then A is orthogonally diagonalizable. G. If A is diagonalizable, then A is invertible. H. If A is diagonalizable, then A is symmetric. I. A is diagonalizable if A = PDP for some diagonal matrix D and some invertible matrix P. J. If A is symmetric, then A is diagonalizable. K. If A is orthogonally diagonalizable, then A is symmetric. L. If there exists a basis for R n consisting entirely of eigenvectors of A, then A is diagonalizable. M. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. EFIJKLM 3. ( pt) UI/Fall/lin span2.pg Which of the following sets of vectors span R 3?. -7, 2 3, , 9-8 C., 2 D. 8 3, -8, E., F.,, -2-7 Which of the following sets of vectors are linearly independent?. -7, 2 3, , 9-8 C., 2 D. 8 3, -8, E., F. B -8, -2-3, ( pt) UI/orthog.pg All vectors and subspaces are in R n. Check the true statements below:. If A is symmetric, Av = rv, Aw = sw and r s, then v w =.. If W = Span{x,x 2,x 3 } and if {v,v 2,v 3 } is an orthonormal set in W, then {v,v 2,v 3 } is an orthonormal basis for W. C. If Av = rv and Aw = sw and r s, then v w =. D. If x is not in a subspace W, then x proj W (x) is not zero. E. If {v,v 2,v 3 } is an orthonormal set, then the set {v,v 2,v 3 } is linearly independent. F. In a QR factorization, say A = QR (when A has linearly independent columns), the columns of Q form an orthonormal basis for the column space of A. G. If v and w are both eigenvectors of A and if A is symmetric, then v w =. BDEF 3. ( pt) local/library/ui/2.3.9.pg Let u be a linear combination of {u,u 2,u 3 }. Select the best statement.. {u,u 2,u 3,u } could be a linearly dependent or linearly independent set of vectors depending on the vectors chosen.. {u,u 2,u 3,u } is never a linearly independent set of vectors. C. {u,u 2,u 3 } is a linearly dependent set of vectors. D. {u,u 2,u 3,u } is always a linearly independent set of vectors. E. {u,u 2,u 3 } is a linearly dependent set of vectors unless one of {u,u 2,u 3 } is the zero vector. F. {u,u 2,u 3 } is never a linearly dependent set of vectors. G. none of the above If u = a u + a 2 u 2 + a 3 u 3, then = a u + a 2 u 2 + a 3 u 3 u {u,u 2,u 3,u } is never a linearly independent set of vectors.

8 37. ( pt) local/library/ui/fall/hw7.pg If A is an n n matrix and b in R n, then consider the set of solutions to Ax = b. Select true or false for each statement. The set contains the zero vector. True This set is closed under vector addition. True This set is closed under scalar multiplications. True This set is a subspace. True A = b, so the zero vector is not in the set and it is not a subspace. 38. ( pt) local/library/ui/fall/hw7.pg Find allvalues of x for which rank(a) = 2. A = 7 2 x x =. - : Row reduce A to get: x x 7 2 Since two pivots are needed, x = 2 G 39. ( pt) local/library/ui/fall/hw8 7.pg Suppose that a matrix A with rows v, v 2, v 3, and v has determinant deta =. Find the following determinants: B = C = D = v v 2 9v 3 v det(b) = C. -2 D. 5 E. -9 F. G. 9 H. 2 I. 5 J. 8 K. None of those above v v 3 v 2 v det(d) = det(c) =. -8. C. -9 D. -3 F. 3 G. 9 H. 2 I. 8 J. None of those above v + 3v 3 v 2 v 3 v. -8. C. -9 D. -3 F. 3

9 G. 9 H. 2 I. 8 J. None of those above D A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution.. True Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.. True G..333 H..5 I. None of those above 3. A + 9I n, eigenvalue = C. -5 D. E. 2 F. G. H. None of those above. A, eigenvalue =. - 8 D. -3 E. -2 F. I. None of those above 2. ( pt) local/library/ui/fall/quiz2 9.pg Supppose A is an invertible n n matrix and v is an eigenvector of A with associated eigenvalue 5. Convince yourself that v is an eigenvector of the following matrices, and find the associated eigenvalues:. A, eigenvalue =.. 8 C. 25 D. 2 E. 2 F. 25 H. None of those above 2. A, eigenvalue = C. -.2 D F F C F D 3. ( pt) local/library/ui/fall/quiz2.pg 3-2 If v = and v - 2 = - are eigenvectors of a matrix A corresponding to the eigenvalues λ = and λ 2 =, respectively, then a. A(v + v 2 ) = C. 5 D. 2 E. F. 3 G. None of those above

10 b. A(3v ) = C. D. 9 E F. G. 3 H. None of those above F E C. - 7 D. E F. 2 8 G H. None of those above D E Suppose a coefficient matrix A contains a pivot in every row. Then A x = b has. (pt) local/library/ui/fall/quiz2.pg Let v = -2, v 2 = -3, and v 3 = be eigenvectors of the matrix A which correspond to the eigenvalues λ = 3, λ 2 = 2, and λ 3 =, respectively, and let v = Express v as a linear combination of v, v 2, and v 3, and find Av.. If v = c v + c 2 v 2 + c 3 v 3, then (c, c 2, c 3 )= 2. Av =. (,2,2). (-3,2,) C. (-,7,3) D. (-2,,2) E. (,,2) F. (,-,5) G. None of above No solution. Unique solution C. Infinitely many solutions D. at most one solution E. either no solution or an infinite number of solutions F. either a unique solution or an infinite number of solutions G. no solution, a unique solution or an infinite number of solutions, depending on the system of equations H. none of the above F Suppose a coefficient matrix A contains a pivot in every column. Then A x = b has. No solution. Unique solution C. Infinitely many solutions D. at most one solution E. either no solution or an infinite number of solutions F. either a unique solution or an infinite number of solutions G. no solution, a unique solution or an infinite number of solutions, depending on the system of equations H. none of the above D

11 7. ( pt) local/library/ui/matrixalgebra/euclidean/2.3.3.pg Let A be a matrix with linearly independent columns. Select the best statement.. The equation Ax = never has nontrivial solutions.. The equation Ax = has nontrivial solutions precisely when it is a square matrix. C. There is insufficient information to determine if such an equation has nontrivial solutions. D. The equation Ax = has nontrivial solutions precisely when it has more columns than rows. E. The equation Ax = has nontrivial solutions precisely when it has more rows than columns. F. The equation Ax = always has nontrivial solutions. G. none of the above The linear independence of the columns does not change with row reduction. Since the columns are linearly independent, after row reduction, each column contains a leading. We get nontrivial solutions when we have columns without a leading in the row reduced matrix. The equation Ax = never has nontrivial solutions. 8. ( pt) local/library/ui/eigentf.pg A is n n an matrices. Check the true statements below:. is an eigenvalue of A if and only if Ax = has a nonzero solution. can never be an eigenvalue of A. C. The vector is an eigenvector of A if and only if det(a) = D. The vector can never be an eigenvector of A E. There are an infinite number of eigenvectors that correspond to a particular eigenvalue of A. F. is an eigenvalue of A if and only if the columns of A are linearly dependent. G. is an eigenvalue of A if and only if det(a) = H. A will have at most n eigenvalues. I. The eigenspace corresponding to a particular eigenvalue of A contains an infinite number of vectors. J. A will have at most n eigenvectors. K. The vector is an eigenvector of A if and only if the columns of A are linearly dependent. L. The vector is an eigenvector of A if and only if Ax = has a nonzero solution M. is an eigenvalue of A if and only if Ax = has an infinite number of solutions DEFGHIM If v and v 2 are eigenvectors of A corresponding to eigenvalue λ, then v + 8 v 2 is also an eigenvector of A corresponding to eigenvalue λ when v + 8 v 2 is not.. True Hint: (Instructor hint preview: show the student hint after attempts. The current number of attempts is. ) Is an eigenspace a subspace? Is an eigenspace closed under linear combinations? Also, is v + 8 v 2 nonzero? Use Cramer s rule to solve the following system of equations for x: x 2y = x + y =.. - Let A = Is A = diagonalizable?. yes. no C. none of the above Note that since the matrix is triangular, the eigenvalues are the diagonal elements 3 and 7. Since A is a 3 x 3 matrix, we need 3 linearly independent eigenvectors. Since 7 has algebraic multiplicity, it has geometric multiplicity (the dimension of

12 its eigenspace is ). Thus we can only use one eigenvector from the eigenspace corresponding to eigenvalue 7 to form P. and let P = Thus we need 2 linearly independent eigenvectors from the eigenspace corresponding to the other eigenvalue 3. The eigenvalue 3 has algebraic multiplicity 2. Let E = dimension of the eigenspace corresponding eigenvalue 3. Then E 2. But we can easily see that the Nullspace of A 3I has dimension. Thus we do not have enough linearly independent eigenvectors to form P. Hence A is not diagonalizable. Let A = yes. no C. none of the above. Is A = diagonalizable? Note that since the matrix is triangular, the eigenvalues are the diagonal elements -5 and. Since A is a 3 x 3 matrix, we need 3 linearly independent eigenvectors. Since has algebraic multiplicity, it has geometric multiplicity (the dimension of its eigenspace is ). Thus we can only use one eigenvector from the eigenspace corresponding to eigenvalue to form P. Thus we need 2 linearly independent eigenvectors from the eigenspace corresponding to the other eigenvalue -5. The eigenvalue -5 has algebraic multiplicity 2. Let E = dimension of the eigenspace corresponding eigenvalue -5. Then E 2. But we can easily see that the Nullspace of A + 5I has dimension 2. Suppose A = PDP. Then if d ii are the diagonal entries of D, d =,. - J. 5 Hint: (Instructor hint preview: show the student hint after attempts. The current number of attempts is. ) Use definition of eigenvalue since you know an eigenvector corresponding to eigenvalue d. H Calculate the determinant of. - J. 5 C Suppose A Thus we have 3 linearly independent eigenvectors which we can use to form the square matrix P. Hence A is diagonalizable Let A = is 5 = Then an eigenvalue of A

13 C Suppose u and v are eigenvectors of A with eigenvalue 2 and w is an eigenvector of A with eigenvalue 3. Determine which of the following are eigenvectors of A and their corresponding eigenvalues. (a.) If v an eigenvector of A, determine its eigenvalue. Else state it is not an eigenvector of A.. - J. v need not be an eigenvector of A (b.) If 7u + v an eigenvector of A, determine its eigenvalue. Else state it is not an eigenvector of A.. - J. 7u + v need not be an eigenvector of A (c.) If 7u + w an eigenvector of A, determine its eigenvalue. Else state it is not an eigenvector of A.. - J. 7u + w need not be an eigenvector of A G G J If the characteristic polynomial of A = (λ + ) (λ 7) 2 (λ + ) 2, then the algebraic multiplicity of λ = 7 is.. C. 2 D. 3 or F. or 2 G. or 2 H.,, or 2 I.,, 2, or 3 C If the characteristic polynomial of A = (λ ) 5 (λ + 3) 2 (λ ) 8, then the geometric multiplicity of λ = 3 is.. C. 2 D. 3 or F. or 2 G. or 2 H.,, or 2 I.,, 2, or 3 G Generated by c WeBWorK, Mathematical Association of America 3