Duke University Math 218D-2 Matrices and Vectors Exam I PDF

Duke University Math 218 Matrices and Vector Spaces Exam I Name: NetID: I have adh...

Duke University Math 218 Matrices and Vector Spaces Exam I Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: September 24, 2021 There are 100 points and 8 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam I Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: February 4, 2022 There are 100 points and 8 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam I Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: September 30, 2022 There are 100 points and 6 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam I Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: February 17, 2023 There are 100 points and 7 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam I Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: September 29, 2023 There are 100 points and 6 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam I Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: February 16, 2024 There are 100 points and 5 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218 Matrices and Vector Spaces Exam II Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: October 22, 2021 There are 100 points and 5 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam II Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: March 4, 2022 There are 100 points and 5 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam II Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: October 27, 2023 There are 100 points and 6 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam II Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: March 22, 2024 There are 100 points and 8 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218 Matrices and Vector Spaces Exam III Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: November 19, 2021 There are 100 points and 4 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam III Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: April 8, 2022 There are 100 points and 4 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam III Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: December 2, 2022 There are 100 points and 4 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam III Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: April 21, 2023 There are 100 points and 6 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam III Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: December 1, 2023 There are 100 points and 4 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Duke University Math 218D-2 Matrices and Vectors Exam III Name: NetID: I have adhered to the Duke Community Standard in completing this exam. Signature: April 19, 2024 There are 100 points and 4 problems on this 50-minute exam. Unless otherwise stated, your answers must be supported by clear and coherent work to receive credit. The back of each page of this exam is left blank and may be used for scratch work. Scratch work will not be graded unless it is clearly labeled and requested in the body of the original problem. MATH  Problem 1. The data below depicts the result of multiplying a real 3 → 3 matrix A by one of its eigenvectors v as well as the unfactored characteric polynomial of A.  A  v    6 21 ↑26 5 15 + 20 i 5 ↑20 203 ↑ 4 i =  25  ωA (t) = c3 t3 + c2 t2 + 49 t ↑ 100 5 ↑24 24 3 ↑ 4 i 25 Note that the coe!cients of t3 and t2 in ωA (t) are labeled as c3 and c2. (5 pts) (a) c3 = and c2 = (4 pts) (b) The eigenvector v of A corresponds to the eigenvalue ε = (note that this ε must be nonreal because A is real and v is nonreal). (4 pts) (c) The value of ε from part (b) of this problem only satisfies one of the following properties. Select this property. ↓ gmA (ε) = 3 ↓ amA (ε) = 3 % & ↓ ωA ε = 0 ↓ ε · I3 ↑ A is invertible ↓ ωA (ε) = trace(A) (5 pts) (d ) Only one of the following statements correctly explains why there is no invertible X and no real-symmetric matrix S satisfying the equation A = XSX →1. Select this statement. ↓ A ↔= XSX →1 because A is not Hermitian. ↓ A ↔= XSX →1 because A is not diagonalizable. ↓ A ↔= XSX →1 because A is not real-symmetric. ↓ A ↔= XSX →1 because A has no real eigenvalues. ↓ A ↔= XSX →1 because A has a nonreal eigenvalue.  Problem 2. The following equation is a digaonaliztion of a 4 → 4 matrix A.    X  D   adj(X)  1 2 0 ↑1 2 0 0 0 4 ↑6 ↗ 11    1 0 ↑1 ↑2  0 1 ↑5 4 ↑2 ↗  A = 0 2 0       2 1 ↑2 10 0 2 0 c  2 ↗ 5 9 ↑1 ↑1 0 ↑1 0 0 0 ↑2 1 2 ↑1 1 1 Note that the last matrix in this factorization is X →1 = adj(X) where c is a scalar quantity. Also note that several c entries in adj(X) are missing and marked as ↗. (6 pts) (a) rank(A) = , trace(A) = , and det(A) = (8 pts) (b) ωA (0) = and ωA (3) = (4 pts) (c) Every value of x such that (x · I4 ↑ A)→1 does not exist satisfies |x| =. (8 pts) (d ) Find the correct value of c. Clearly explain your reasoning to receive credit. (8 pts) (e) Is A a real symmetric matrix? Clearly explain why or why not using ideas from class. (16 pts) Problem 3. Let u(t) be the solution to the initial value problem u↑ = Au with u(0) = u0 where ) * ) * 7 6 ↑3 A= u0 = ↑3 ↑2 2 Calculate u(ln(2)). Simplify your answer to a vector of integers. Hint. The algebraic identities a ln(b) = ln(ba ) and eln(a) = a will be useful here.  Problem 4. Consider the following singular value decomposition A = U ”V ↭ (note that A is 5 → 4).  A   ↓ U  V↭ 0  ↘ ! ↑6 6 6 ↑6 3/2 0 0   ↑1 ↓ ↓ ↓ 8 3 ↘ ↑1/2 1/2 1/2 ↑1/2  ↑3 ↑3 ↑5    ↓ 0 3/3 3/3 ↑ 3/3 ↑1/2 ↑2 ↓ ↓ 6 3 ↑1/2 ↑1/2 ↑1/2  0 4 ↑2  3  =  ↓ /6 0 3/3 3/3  ↘   1/2  ↑2 ↓ ↓ 2 3 ↑1/2 1/2 ↑1/2 ↑6 ↑4 0 ↑ 3/6 3/3 0 3/3 ↘ ↑6 0 ↑2 ↑4 ↓ 3/6 ↓ 3/3 ↓ ↑ 3/3 0 2 3 ↑1/2 ↑1/2 1/2 1/2 Throughout this problem, let S = A↭ A and let q(x) be the quadratic form q(x) = ≃x, Sx⇐. (6 pts) (a) rank(A) = and rank(S) = (6 pts) (b) Which categories of definiteness apply to S? Select all that apply. ↓ positive definite ↓ positive semidefinite ↓ negative definite ↓ negative semidefinite ↓ indefinite (4 pts) (c) The smallest real number x such that det(x · I4 ↑ S) = 0 is x =. (8 pts) (d ) Let A+ = V ”→1 U ↭. Calculate AA+ A. Your answer will be a 5 → 4 matrix. + ,↭ (8 pts) (e) Let v be any vector satisfying V ↭ v = 0 0 1 1. Calculate the value of q(v). Simplify your answer to an integer value.  Problem 1. Suppose that A = QR where A, Q, and R are given by     1 1 →1 ↑    5 4 0   1 →1 1 →x A=  a1 a2 a3   Q=    R= 0 2 ↑7    h 1 x 1 0 0 5 x →1 →1 Note that the columns of A have been labeled a1 , a2 , a3 and that the formula for Q depends on variables x and h. (6 pts) (a) rank(A) = , rank(R) = , and rank(Q) = (5 pts) (b) h = (your formula for h here should depend on the variable x) (8 pts) (c) det(R) = , det(RQ↭ Q) = , and det(RA↭ A) = (6 pts) (d ) If q 2 is the second column of Q, then ↓q 2 , a1 ↔ = , ↓q 2 , a2 ↔ = , and ↓q 2 , a3 ↔ = (6 pts) (e) If q 1 is the first column of Q, then only one of the following statements is correct. Select this statement. ↗ projq1 (a1 ) = O ↗ projq1 (a2 ) = O ↗ projq1 (a3 ) = O ↗ none of these equations is correct ' (↭ (10 pts) (f ) Find the projection of b = h2 0 0 0 onto Col(Q) (your answer will deend on the variable x).  Problem 2. The following equation depicts A = XBX →1 , which tells us that A is similar to B.    X  B   i 1 2 →1 5 2 0 9   1 i →1 →i  4    A = 0 7 i  X →1    1 →3 0 →10 0 1 i   1 →1 i 2 0 0 0 i Note that several entries in X and in B are nonreal complex numbers and that B is upper triangular. (4 pts) (a) trace(A) = and det(A) = (4 pts) (b) If x1 is the first column of X and x2 is the second column of X, then ↓x1 , x2 ↔ =. (8 pts) (c) Note that trace(X) = 2i+2. This calculation allows us to decide whether or not each of the following statements is true. Select each true statement (each option is worth 2pts). ↗ ω = 2i + 2 is an eigenvalue of X ↗ X has at least one nonreal eigenvalue ↗ the coe!cient of t3 in εX (t) is 2i + 2 ↗ X cannot be similar to any Hermitian matrix (3 pts) (d ) The algebraic multiplicity of every eigenvalue ω of A is amA (ω) =. (10 pts) (e) Note that ω = 5 and ω = 7 are both eigenvalues of A. Find bases of EA (5) and EA (7) and determine if EA (5) ↘ EA (7). Hint. Start by finding bases of EB (5) and EB (7). How do bases of these eigenspaces then translate into bases of EA (5) and EA (7)?  Problem 3. The data below depicts an invertible real-symmetric matrix S, an invertible matrix T , and the charac- teristic polynomial εS (t) of S (which has been partially factored).     2 →1 1 2 →7 1 →1 →1 →1 2 →1 →2  0 1 2 →1 ) *) * S=  1 →1  T =  εS (t) = t2 → 2 t + 1 t2 → 9 t + 8 2 2 →10 14 1 →2 2 →2 2 5 1 5 →2 1 Throughout this problem, let A = M →1 T where M = S →1 T. (6 pts) (a) Determine the definiteness of S. Clearly explain your reasoning to receive credit. (10 pts) (b) Show that A is similar to S. Hint. This can be done purely with symbols. (14 pts) Problem 4. Suppose that u(t) is the solution to u↑ = Au with u(0) = u0 where + , + , →1 1 3 A= u0 = (a + 1) · 0 a 1 Note that the matrix A and the vector u0 are defined in terms of a real variable a which is known to satisfy a ≃= →1. The two coordintes u1 and u2 of u(t) depend both on t and a and can thus be interpreted as scalar fields. Calculate ϑu1 ϑu2 the partial derivatives and. ϑa ϑa  Problem 1. Consider the matrix L given by     3 1 2 4 5 4 → 0 ↑4 → 0 1 2 3 4 → 12 → 24 ↑54     L= 0 0 2 5 9  adj(L) =  0 → 6 ↑30 78  0 0 0 1 7 0 → 0 12 ↑42 0 0 0 0 2 → 0 0 0 6 Note that several entries in adj(L) are unknown and marked →. (4 pts) (a) The trace of L adj(L) is. (4 pts) (b) The (2, 4) entry of L→1 is. (5 pts) (c) The missing (5, 1) entry of adj(L) is. Show your work below to receive credit, but fill in the blank to make your answer clear. (5 pts) Problem 2. All of the roots of the polynomial f (t) below are known to be integers. f (t) = t9 ↑ 15 t8 + 81 t7 ↑ 179 t6 + 75 t5 + 303 t4 ↑ 397 t3 ↑ 9 t2 + 240 t ↑ 100 One of the following numbers is not a root of f (t). Select this number. ↓ ↑1 ↓ 1 ↓ 2 ↓ 3 ↓ 5 Problem 3. Suppose that A is an n ↔ n matrix whose characteristic polynomial is given by ωA (t) = t10 ↑ 19 t9 + 123 t8 ↑ 266 t7 + 13 t6 ↑ 36 t5 ↑ 96 t4 + 171 t3 ↑ 24 t2 ↑ 120 t + 53 It is known that ε1 = 7 + 2i is an eigenvalue of A. (4 pts) (a) The trace of A is and the determinant of A is. (4 pts) (b) An eigenvalue ε2 ↗= ε1 of A is ε2 = (any eigenvalue of A not equal to ε1 is valid here). (8 pts) (c) Which, if any, of the following statements is correct? Select all that apply (two points each). ↓ A is nonsingular ↓ A is Hermitian ↓ A is real symmetric ↓ A adj(A) is positive definite    ↑1 1 4  (20 pts) Problem 4. Determine if A = ↑3 3 3 is diagonalizable. Clearly explain your reasoning to receive credit. ↑1 1 4  Problem 5. The following equation depicts a spectral factorization of a real symmetric matrix S.  U  D  U↭    2/3 ↑1/3 0 2/3 7 2/3 ↑1/3 ↑2/3 0  1 0  ↑1/3 ↑2/3  S  = ↑ /3 ↑2/3 ↑2/3  ↑2  ↑2/3 0  ↑2/3 0 c 2/3 bi  0 ↑2/3 c 2/3 0 ↑2/3 2/3 ↑1/3 ↑bi 2/3 0 2/3 ↑1/3 Note that the variable b in D and the variable c in U (and U ↭ ) are currently unknown. (9 pts) (a) The trace of S is , the value of b is , and the value of c is. (4 pts) (b) What is the definiteness of S? Select all that apply. ↓ positive definite ↓ positive semidefinite ↓ negative definite ↓ negative semidefinite ↓ indefinite (4 pts) (c) The largest eigenvalue of S is and the smallest eigenvalue of exp(S) is. (10 pts) (d ) Let q be the quadratic form defined by S. Calculate q(3, 6, 0, 6).  Problem 6. The equations below depict a matrix A and a spectral factorization of A↭ A.    V   V↭  1 ↑1 ↑1 ↑1 1/→6 1/→6 1/→3 1/→3  D  1/→6 ↑ 2/→6 0 ↑ 1/→6 0 10  0 0 1  ↑ 2/→6 0 0 1/→3 → 1/ 6 0 2/→6 1/→6 8 A= 0 ↑1 2 ↑1 A↭ A =   2/→6 ↑ 1/→3   → ↑ 1/→3 1/→3  2 ↑1 1 1  0 0 5 1/ 3 0 ↑ 1/→6 1/→6 1/→3 ↑ 1/→3 1 1/→3 1/→3 0 ↑ 1/→3 0 2 1 1 (3 pts) (a) The rank of A is r =. (6 pts) (b) The largest singular value of A is ϑ1 = and the smallest singular value of A is ϑr =. (10 pts) (c) Suppose we use the given spectral factorization of A↭ A to calculate singular value decomposition A = U !V ↑. Find the last column of U.  Problem 1. Consider the matrix A and its characteristic polynomial ωA (t) given by   1 27 0 0 0 0 →1 0 0 0   A=  0 0 2 1 2  ωA (t) = t5 + t 4 → 5 t 3 + 5 t2 + 4 t → 4 0 0 8 3 c 0 0 →4 →1 →4 Note that the (4, 5) entry of A is a variable marked c. Also note that the coe!cient of t4 in ωA (t) is blank. (4 pts) (a) Fill in the blank coe!cient of t4 in ωA (t). (10 pts) (b) Find c. Hint. What is det(A)? (10 pts) (c) The scalar 2 is an eigenvalue of A. Suppose that v is an eigenvector of A corresponding to the eigenvalue 2. Show that v is also an eigenvector of adj(A) and identify its corresponding eigenvalue ε.  Problem 2. Consider the following matrix factorization A = XDX →1.    X  D  X →1  1 →3 0 1 3 0 0 0 →7 →9 →2 2    0 1 0 →1  0  →4 →5 →1 1  A = 0 i 0     →2 2 1 1 0  0 1→i 0 →2 →2 0 1 2 →4 1 0 0 0 0 1 → i →4 →6 →1 1 (5 pts) (a) The eigenvalue of A with the largest geometric multiplicity is ε = and that multiplicity is.    →3   1 (4 pts) (b)   A   =  2 →4 (8 pts) (c) Calculate det(A). Simplify your answer to a complex number of the form a + b i. (8 pts) (d ) Calculate ωA (3 → i). Simplify your answer to a complex number of the form a + b i.  Problem 3. Consider the factorization A = XBX →1 below.  A   X  B  X →1  →4 2 →7 1 2 2 0 1 2 →3 2 →4 →2 1 →5 = 2 1 →10 0 1 4 →2 5 B2 = B3 = 2 →1 3 0 →1 →2 0 0 0 →2 1 →3 (3 pts) (a) Fill in the blanks above to calculate B 2 and B 3. (4 pts) (b) trace(A) = , det(A) = , and ωA (t) =   1 t 1/2 t2 + 2t  (10 pts) (c) Use the Taylor series definition of matrix exponentials to show that exp(Bt) = 0 1 t. 0 0 1 ' (↭ (10 pts) (d ) Let u(t) be the solution to u↑ = Au with initial condition u(0) = 1 1 0. Calculate u(2).  Problem 4. Consider the quadratic form q(x) = ↑x, Sx↓ where S is the singular real-symmetric matrix satisfying  v1     v 2     v3   0 0 →1 →3 1 12  1 3   3  9  0  0   S   =   0 0   S   =   →1 →3   S   =   1 12 0 0 c 3c 2 24 Note that the last coordinate of the vector v 2 above is marked as the variable c. (7 pts) (a) The trace of S is trace(S) = and the value of c is c =. (3 pts) (b) Which of the following adjectives apply to q(x)? Select all that apply (no partial credit on this problem). ↔ positive definite ↔ positive semidefinite ↔ negative definite ↔ negative semidefinite ↔ indefinite (4 pts) (c) Which of the following vectors x satisfies q(x) = 0? ' (↭ ' (↭ ' (↭ ↔ x = 1 0 →1 0 ↔ x = 1 1 →1 0 ↔ x= 0 3 →2 1 ↔ none of these (6 pts) (d ) Complete the square to write q(x) as a linear combination of squares. Note. You may leave c as a variable to solve this problem. ϑq (4 pts) (e) The quadratic form q(x) = q(x1 , x2 , x3 , x4 ) is a scalar field on R4. Calculate. ϑx2  Problem 1. Suppose that w1 , w2 , w3 → R4 are mutually orthogonal vectors satisfying ↑w1 ↑ = ↑w2 ↑ = ↑w3 ↑ = c where c > 0. Further suppose that A factors as A = W R where     | | | 1 1 1 W = w 1 w 2 w 3  R = 0 1 1 | | | 0 0 1 (5 pts) (a) Find W ↭ W (note that this matrix depends on the scalar c). (5 pts) (b) Show that A↭ A = c2 · R↭ R. % &↭ (8 pts) (c) Suppose that b is a vector satisfying W ↭ b = c4 · 1 ↓1 1. Find the least squares approximate solution x ' ' depends on the scalar c). to Ax = b (note that x (7 pts) (d ) Is R diagonalizable? Explain why or why not.    1 i i 0 ↓1 ↓3i + 1 1 1 Problem 2. Consider the nonsingular matrix A =  ↓1 . ↓i ↓i i ↓1 ↓i 4i 7 (13 pts) (a) Find det(A). (12 pts) (b) Find the (1, 4) entry of det(A) · A→1.  Problem 3. Suppose that A is a matrix whose characteristic polynomial is given by ωA (t) = t6 ↓ 6 t4 ↓ 4 t3 + 9 t2 + 12 t + 4 (8 pts) (a) trace(A) = and det(A) = →1 (9 pts) (b) Does (I ↓ A) exist? Clearly explain why or why not. (8 pts) (c) If possible, find rank(A). If this is not possible then explain why.  Problem 4. Suppose that A = XDX →1 where       ↔ ↓235 71 ↓237 1 1 8 ↓24 2 0 0 0 ↔ 19 ↓4 12   0 1 1 ↓4 0 3 0 0 A= ↔  X=  D=  435 ↓112 357 ↓1 9 15 ↓65 0 0 ↓1 0 ↔ 40 ↓8 23 0 2 6 ↓23 0 0 0 ↓1 Note that the first column of A is currently unknown. (5 pts) (a) Find the complete factorization of ωA (t). Clearly explain your reasoning to receive credit. (10 pts) (b) Find the missing column of A. Clearly explain your reasoning to receive credit. % &↭ Hint. Note that v = 1 0 ↓1 0 is the first column of X. % &↭ (10 pts) (c) Suppose that u0 → R4 satisfies X →1 u0 = 1 0 1 0 and that u(t) solves the initial value problem u↑ = Au with u(0) = u0. Which, if any, of the coordinates of u(t) tend to zero as t ↗ ↘? Clearly explain your reasoning to receive credit.  Problem 1. Consider the invertible matrix A and its cofactor matrix C given by 2 3 2 3 1 1 1 0 2 1 1 1 2 3 6 1 0 1 1 07 6 3 0 3 3 07 6 7 6 7 A=6 6 1 1 0 1 07 7 C = 6 3 6 ⇤ ⇤ 3 37 7 4 2 1 2 0 15 4 2 1 ⇤ ⇤ 05 0 0 1 1 1 0 3 3 3 3 Note that C is missing several entries. (10 pts) (a) Find the missing (4, 3) entry of C. (10 pts) (b) Find A (0) (the constant coefficient of the characteristic polynomial of A). Hint. This can be done by calculating a single inner product. ⇥ ⇤| (10 pts) (c) Find the solution x to the system Ax = b where b = det(A) · 1 0 0 0 1. (20 pts) Problem 2. Consider the system of di↵erential equations given by f0 = 5f 3g f (0) = 1 g0 = 6f 4g g(0) = 2 Find f (t) and g(t).  Problem 3. The matrix H below is Hermitian with exactly two eigenvalues E-Vals(H) = { 1 , 2} where 1 =2 and 2 is unknown. A basis of EH (2) is given below. 2 3 8 ⇤ 2 ⇤ 6 ⇤ 8 ⇤ ⇤7 ⇥ ⇤| H=6 4 2 ⇤ 7 EH (2) = Span{ i i i 1 } 8 ⇤5 ⇤ ⇤ 2i 8 Note that H is missing several entries. (3 pts) (a) The (3, 4) entry of H is. (3 pts) (b) The algebraic multiplicity of 1 = 2 as an eigenvalue of H is. (3 pts) (c) The algebraic multiplicity of 2 as an eigenvalue of H is. (3 pts) (d ) The coefficient of t3 in H (t) is. ⇥ ⇤| (8 pts) (e) Determine if i 1 i i 2 EH ( 2 ). Explain your resoning. (8 pts) (f ) Find 2. Explain your reasoning.  Problem 4. Consider the quadratic form on R3 given by q(x1 , x2 , x3 ) = (x1 + 4 x2 + 5 x3 )2 + (3 x1 2 x2 + x3 ) 2 + ( 2 x1 + x2 x3 )2 + (2 x1 + 5 x2 + 7 x3 )2 Note that this quadratic form may be written as q(x) = hx, Sxi where S is real-symmetric. (3 pts) (a) Which of the following adjectives correctly describes q(x)? positive semidefinite negative semidefinite indefinite (3 pts) (b) Which of the following correctly describes the eigenvalues of S? Some eigenvalues of S are positive and some eigenvalues of S are negative. The eigenvalues of S are all nonpositive. The eigenvalues of S are all nonnegative. (10 pts) (c) If possible, find A such that S = A| A. If this is not possible, then explain why. (6 pts) (d ) If possible, find x 6= O satisfying q(x) = 0. If this is not possible, then explain why.  Problem 1. The two equations below depict matrix-vector products Av and A↭ w (note that this matrix A is 3 → 4).   v      ↑t + 4   w 2s ↑ 2  ↑t + 5 ↑t + 2   2 s    A   =  2 t ↑ 4  A↭ ↑s =  s ↑ 1   3   ↑3 s + 3 4t ↑ 8 1 ↑t + 4 s↑1 Note that the formula for v (and thus the formula for Av) depends on a scalar t and that the formula for w (and thus the formula for Aw) depends on a scalar s. (5 pts) (a) Every vector in Null(A) has coordinates and every vector in Col(A) has coordinates. (5 pts) (b) Setting t = 2 in the equation for Av above leads us to which of the following conclusions?         2 2 2 2 3  3 3 ↭ 3 ↭ ↓   3 ↔ Null(A) ↓   3 ↔ Col(A) ↓   3 ↔ Col(A ) ↓   3 ↔ Null(A ) 2 2 2 2 (5 pts) (c) Setting t = 0 in the equation for Av above leads us to which of the following conclusions?     4 4       5 5 2 2 2 ↓   3 ↔ Col(A) ↓   3 ↔ Null(A) ↓ ↑4 ↔ Null(A) ↓ ↑4 ↔ Col(A) ↓ ↑4 ↔ Col(A↭ ) ↑8 ↑8 ↑8 4 4 (5 pts) (d ) Setting s = 1 in the equation for A↭ w above leads us to which of the following conclusions?         2 2 2 2 ↓ ↑1 ↗ Null(A) ↓ ↑1 ↗ Col(A↭ ) ↓ ↑1 ↗ Null(A↭ ) ↓ ↑1 ↗ Col(A) 1 1 1 1 (5 pts) (e) Setting s = 0 in the equation for A↭ w above leads us to which of the following conclusions?        ↑2   ↑2 ↑2 0 ↑1 0 ↑1 ↑1 ↭ ↓ 0 ↗ Col(A) ↓    3 ↗ Col(A) ↓ 0 ↗ Null(A) ↓    3 ↗ Null(A) ↓    3 ↗ Col(A ) 1 1 ↑1 ↑1 ↑1 (14 pts) Problem 2. Suppose that A is 9 → 4 with linearly independent columns. Fill in every missing label in the picture of the four fundamental subspaces of A below, including the dimension of each fundamental subspace. A R R Problem 3. The equation below defines a matrix A = BU ↭ Q↭. Here, B is a 1 → 3 matrix, U is a 4 → 3 matrix, and Q is a 5 → 4 matrix. It is known that both U and Q have orthonormal columns.  Q↭   U↭  437/537 250/537 14/537 16/179 60/179 19/121 46/121 10/11 ↑8/121  54 ' B ( 34/121 ↑ /179 135/179 72/179 ↑60/179 ↑46/179 A= 3 5 1  10/121 107/121 ↑4/11 ↑250/537 88/537 35/537 40/179 150/179 118/121 ↑20/121 ↑1/11 14/121 40/537 ↑100/537 424/537 101/179 ↑24/179 (6 pts) (a) Q↭ Q = and U ↭ U = (use notation that conveys the size of these matrices for full credit). (8 pts) (b) Calculate AA↭ (this will be a scalar value since A is 1 → 5). Hint. Most of this calculation should involve symbolic manipulation (there is no hope of carrying out this calculation numerically). (8 pts) Problem 4. The expression below depicts a sequence of row reductions applied to a 4 → 4 matrix A. It is known that det(A) = ↑30.        L  1 0 0 0 + 3·r 1 → r 2   rr24 ↑ 15·r 1 → r 4   r2 ↓r3   r4 +7·r2 →r4 ≃ 2 0 0  A  ↑↑↑↑↑ ↑↑↑↑↑↑↘  A1  ↑↑↑↑↘  A2  ↑↑↑↑↑↑↑↑↘         ≃ ≃ 3 0 ≃ ≃ ≃ h Note that the lower-triangular matrix labeled L is missing several entries marked ≃ and has a variable h in its (4, 4) position. Find the correct value of h. Clearly explain your work to receive credit. Problem 5. The data below depicts a matrix A and the result of projecting a vector b onto Col(A).       1 0 ↑2 ↑3  1 2 ↑2 ↑1       A=  0 1  Projection of b =      0 onto Col(A) is  1.  0 1  4  1 0 1 1 1 In this problem, P is the projection matrix onto Col(A). (5 pts) (a) trace(P 2 ) =         ↑5 1 5 ↑1 1 ↑3 ↑1  3  1 ↑1           (5 pts) (b) What is the projection of b onto Null(A↭ )? ↓   1  ↓   ↑1 ↓  ↑1  ↓   1  ↓    1  5  3 ↑5 ↑3  1 2 0 ↑2 0 0 1 ' (↭ (8 pts) (c) Using the Gram-Schmidt algorithm to calculate Q in A = QR produces q 1 = ⇐ 1 1 0 0 0 as the first 2 column of Q. Find the second column q 2 of Q. Clearly show your work (including any formulas you use) to receive credit.  (5 pts) Problem 6. Consider the three data points {(3, 7), (2, ↑5), (↑6, 9)}. The fact that these three data points do not all lie on a line is equivalent to the inconsistency of which of the following augmented systems?       ) * ) * 3 7 0 1 3 7 9 49 1 3 ↑1 11 1 1 1 11 ↓ ↓ ↓  2 ↑5 0 ↓ 1 2 ↑5 ↓  4 25 1 ↑1 49 ↑43 3 2 ↑6 ↑43 ↑6 9 0 1 ↑6 9 36 81 1 (6 pts) Problem 7. Consider the three data points {(↑1, 1), (1, 2), (2, 2)}. The fact that it is impossible to fit a curve of the form c1 (x2 ↑ y) + c2 (x ↑ y 2 ) = 1 to these data points is equivalent to the inconsistency of which of the following augmented systems?         0 ↑2 1 ) * 1 ↑1 1 0 2 1 0 2 1 5 ↑1 1 ↓ ↑1 ↑3 1 ↓ ↓ 1 1 2 ↓ ↑1 5 1 ↓ ↑1 5 1 ↑1 17 ↑7 2 ↑2 1 1 2 2 2 8 1 2 8 1 (10 pts) Problem 8. It is impossible to fit the three data points {(1, ↑1) , (1, ↑2) , (↑1, 1) } to a curve of the form c1 x3 + c2 (x ↑ y) = 2 We can, however, use the method of least squares to approximate these data points and this process produces the curve ↑4 x3 + 2 (x ↑ y) = 2 (so + c1 = ↑4 and + c2 = 2). Find the error E in using this method to approximate this data. Hint. You will earn four points for identifying the the system Ax = b whose inconsistency is equivalent to the fact that we cannot perfectly fit to the original curve to the data.  Problem 1. Each of the three equations below provides information about the eigenvalues of A (the matrix in the first of the three equations).  A  v 1     v2       2 →3 5 →6 3 15 1 3 →5 6 2 0 2 3 →5 6 1 0 →1 0  1 0 3 →8  4  20 →1 3 →3 8 ↑ 0 →1 4 →3 8 0 1 →1 2    =      =   rref  =  →1 →3 8 →6 3  15  1 3 →5 6↑ 0  1 3 →4 6 0 0 0 0 →1 1 0 6 →1 →5 1 →1 0 →3 ↑ 0 1 →1 0 →2 0 0 0 0 Note that the vector v 2 from the second equation is missing three entries marked ↑. (4 pts) (a) The vector v 1 is an eigenvector of A corresponding to the eigenvalue ω =. (5 pts) (b) Only one of the following statements correctly characterizes the vector v 2. Select this statement. ↓ v 2 ↔ EA (3) ↓ v 2 ↔ EA (→3) ↓ v 2 ↔ EA (2) ↓ v 2 ↔ EA (→2) ↓ v 2 ↔ / EA (ω) for any value of ω (5 pts) (c) The eigenvalue ω = of A satisfies gmA (ω) = 2. (4 pts) (d ) Which of the following statements correctly summarizes the data of the eigenvalues of A? ↓ A has two eigenvalues with geometric multiplicity two. ↓ A has two eigenvalues with geometric multiplicity one and one eigenvalue with geometric multiplicity two. ↓ A has one eigenvalue with geometric multiplicity one and two eigenvalues with geometric multiplicity two. ↓ A has three eigenvalues with geometric multiplicity two. ↓ A has three eigenvalues with geometric multiplicity one. (10 pts) Problem 2. Suppose that G is a directed graph with exactly one connected component and that A is the incidence matrix of G. The equation below depicts the result of Brian’s attempt to multiply A by a matrix X and record the result as Y.  Y    X  1 0 0 2 2 0 2 0   0 1 2 0  0 0 →1 0     →3 0 →2 →2    1 1 0 2         A 0 1 1 2 =  0 0 →1 →2     2 0 0 0    0 2 2 0    0 0 1 0 0 1 1 0 0 0 3 2 Despite his best e!orts, Brian made a mistake in his calculation! One (and only one) of the columns of Y above is incorrect. Identify which column of Y was calculated incorrectly. Clearly explain your reasoning to receive credit. (12 pts) Problem 3. Select every matrix below whose columns are linearly independent (each option is worth 2pts).       0 1 0 0 5   3 1 3 3 3 1 2 4 2 1 3 2 0 0 1 0 8 1 1 2 3 2 1 1   2 2  2 4 2 4  2 6  4  ↓     0 0 0 1 8 ↓ 3 0 ↓ 4 1 3 3 1  4 1 1 4   ↓ 1 3 2  0 0 0 0 0 3 4 4 4 2 1 3 2 3  2 6 4 4 4 0 0 0 0 0 1 4 1 3 3 4 4 2 4 1 3 2   4 24 9 37 6 7 131 4 4 6   0 0 3 3 9 5 86 4 1 8 1 1 6 3   0 0 0 0 6 1 43 2 2 4 0 5 8 9     0 0 0 0 0 8 8 8 2 1 0 0 4 3  ↓     0 0 0 0 0 0 0 9 6 2 ↓ 0 0 0 7     0 0 0 0 0 0 0 0 8 6 0 0 0 0   0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (12 pts) Problem 4. Suppose that A is a 9 ↗ 7 matrix and that v ↔ R7 and w ↔ R9 satisfy the two equations Av = 9 · w A↭ w = 5 · v where v ↘= O. Show that v is an eigenvector of the Gramian of A and identify the corresponding eigenvalue.  Problem 5. Suppose that A is a matrix satisfying the following properties              5 9 3  1 2   0 1 8     2 h 0 0 2         ,   ↔ Null(A) 0  0 Null(A↭ ) = Span  ,  ,   0 0 0         1 2   0 0 0       0 0 0  Note that the second vector in the first equation above is defined in terms of a constant labeled h. (10 pts) (a) Fill in every missing label in the picture of the four fundamental subspaces of A below, including the dimension of each fundamental subspace. A R R (4 pts) (b) The only valid value of h is h =. (4 pts) (c) Only one of the following formulas for b makes the system Ax = b consistent. Select this vector.           1 0 0 0 1  0 1 0  0 1            0 0 1  0 1 ↓ b=  ↓ b=  ↓ b=  ↓ b=  ↓ b=          1 1   0   0   1      0 1 0  1 0 0 0 1 1 1 (8 pts) (d ) Find the projection matrix onto Null(A).  Problem 6. The data below depicts a matrix A and the result of multiplying the projection matrix onto Col(A) by another matrix Y.      Y    ↑ 1 3   →3 24 3 5 4 0 23 4 5 3   ↑ ↑ 0   7 →3 →29 0 →1 4 →2 →30 0 0 A= 2 1 →1    PCol(A)   7 =    16 →29 1 →2 4 17 →30 1 →1 1 ↑ →1   →10 2 →20 →1 3 2 →2 →16 →1 →1 Note that A is missing several entries marked ↑. (6 pts) (a) Only one of the columns of Y is in the column space of A. Select this column. ↓ first column ↓ second column ↓ third column ↓ fourth column ↓ fifth column. /↭ (9 pts) (b) Let b = 4 →1 →2 3 (the last column of Y ). Find the least squares approximate solution x  to Ax = b.. /↭ (7 pts) (c) Let b = 24 →3 16 2 (the second column of Y ) and let x  be the least squares approximate solution to Ax = b. Find the error E in this approximation.  Problem 1. The following row-reductions calculate the matrix U in P A = LU.  A       U  6 →36 7 r2 → 2·r1 ↑ r2 6 →36 7 6 →36 7 6 →36 7 r 3 + 3·r 1 ↑ r 3  12 →72 14 r 4 + r 1 ↑ r 4 0 0 0 r ↓r  3 0 0 5 r 4 → 4·r 2 ↑ r 4 0 0 5  → →→→→→→→→→↑   →→2 →→↑  →→→→→→→→→→↑   →18 108 →16 0 0 5 0 0 0 0 0 0 →6 36 13 0 0 20 0 0 20 0 0 0 (7 pts) (a) Find L. ' (↭ (10 pts) (b) Suppose that b is a vector satisfying 60 →60 0 0 = L→1 P b. Use this information to find all solutions x to Ax = b. Clearly explain your reasoning to receive credit. (4 pts) (c) Which of the following statements accurately describes the rows and columns of A? ↓ The rows of A are independent and the columns of A are independent. ↓ The rows of A are dependent and the columns of A are independent. ↓ The rows of A are independent and the columns of A are dependent. ↓ The rows of A are dependent and the columns of A are dependent. (4 pts) (d ) The last two rows of one of the following matrices form a basis of Null(A↭ ). Select this matrix. ↓ L→1 P ↓ L ↓ LP ↓ P L→1 ↓ U L  Problem 2. Suppose that A is a 3 ↔ 3 matrix satisfying the following three equations.              1 1 1 1 0 1 0 1 1 0 1 →11  A 1 = →2 rref  A 1 = 0 1 0 0 rref  A↭ 1 =  0 1 1 →7 1 →1 1 0 0 0 1 1 0 0 0 0 Note that rref(A) and rref(A↭ ) can be inferred from the second and third equations above. ' (↭ (4 pts) (a) The vector 1 1 1 belongs to exactly one of the four fundamental subspaces of A. Select this space. ↓ The null space. ↓ The row space. ↓ The column space. ↓ The left null space. ' (↭ (4 pts) (b) The vector 1 →2 →1 belongs to exactly one of the four fundamental subspaces of A. Select this space. ↓ The null space. ↓ The row space. ↓ The column space. ↓ The left null space. ' (↭ (5 pts) (c) Determine if 1 →1 1 ↗ Null(A). Clearly explain your reasoning to receive credit. (5 pts) (d ) Find a basis of the left null space of A. Clearly explain your reasoning to receive credit. (7 pts) Problem 3. Suppose that A and B are n ↔ n matrices and that v ↗ Rn satisfies v ↗ EA (→2) and v ↗ EB (5). Show that v is an eigenvector of M = A2 + AB → In and identify the corresponding eigenvalue.  Problem 4. Suppose that A is a matrix whose null space and left null space are given by  K  1 0 0 3  C   1 1 0  2 0 7  0 ↭ 1 Null(A) = Col   2 3 4 32 Null(A ) = Col  →1   0 1 0 1 5 →1 1 →1 →1 →2 →15 Note here that K is 5 ↔ 4 and that C is 4 ↔ 2. ' (↭ ' (↭ (6 pts) (a) Explain why { 1 0 →1 →1 , 0 1 1 1 } is a basis of Null(A↭ ). ' (↭ (6 pts) (b) Find all values of c for which Ax = 2 c →3 5 is consistent. Clearly explain your reasoning to receive credit. (10 pts) (c) Fill in every missing label in the picture of the four fundamental subspaces below, including the dimension of each fundamental subspace. A R R (5 pts) (d ) Does K have independent or dependent columns? Clearly explain your reasoning to receive credit.  Problem 5. Suppose that A is a 4 ↔ 3 matrix and that the projection matrix P onto Col(A) satisfies the following equations.                   →1 →1   0 0   →1 →1   2 2 →3 →3 0 0 →8 →8  1 1  P   =    P   =    P   =    P   =   →1 →1 0 0  0  0  1 1 0 0 1 0 0 0 →4 0 (6 pts) (a) Which of the following vectors belongs to EP (1)? Select all that apply. ' (↭ ' (↭ ' (↭ ' (↭ ↓ →1 →3 →1 0 ↓ 0 0 0 1 ↓ →1 →8 0 0 ↓ 2 1 1 →4 (6 pts) (b) Which of the following vectors belongs to Null(A↭ )? Select all that apply. ' (↭ ' (↭ ' (↭ ' (↭ ↓ →1 →3 →1 0 ↓ 0 0 0 1 ↓ →1 →8 0 0 ↓ 2 1 1 →4 ' (↭ (6 pts) (c) Find the projection of b = 2 1 1 →4 onto Null(A↭ ). Clearly explain your reasoning to receive credit. (5 pts) (d ) Find' the error E in( using the technique of least squares to approximate a solution to the system Ax = b where ↭ b = 2 1 1 →4. Clearly explain your reasoning to receive credit.  2 3 1 0 2 Problem 1. Consider the matrix A = 4 1 1 15. 1 0 2 ⇥ ⇤| (6 pts) (a) Determine if 1 1 0 2 Null(A). ⇥ ⇤| (6 pts) (b) Determine if 1 0 1 2 Col(A). ⇥ ⇤| (6 pts) (c) Is 1 1 1 an eigenvector of A? If so, what is the associated eigenvalue? (7 pts) Problem 2. Suppose that A is a matrix and v is a vector satisfying v 2 EA ( 3). Show that v is an eigenvector of M = A2 A and identify the corresponding eigenvalue.  ⇥ ⇤| ⇥ ⇤| Problem 3. The first row of a matrix A is the vector 1 1 0 1 and the first column is the vector 1 3 1. ⇥ ⇤| (7 pts) (a) If possible, determine if 2 4 3 2 Null(A| ). If this is not possible, then explain why. (10 pts) (b) Now, suppose that Null(A) = Span{v 1 , v 2 , v 3 } where {v 1 , v 2 , v 3 } is linearly independent. Fill in every missing label in the picture of the four fundamental subspaces below, including the dimension of each fundamental subspace. A R R (8 pts) (c) Suppose again that Null(A) = Span{v 1 , v 2 , v 3 } where {v 1 , v 2 , v 3 } is linearly independent. Find A.  Problem 4. A matrix A has projection onto Col(A| ) and projection onto Col(A) given by 2 3 2/3 1/3 1/3 0 2 3 5/9 4/9 2/9 6 1/3 1/3 0 1/37 PCol(A| ) = 6 4 1/3 1/35 7 PCol(A) = 4 4/9 5/9 2/95 0 1/3 2/9 2/9 ⇤ 0 1/3 1/3 2/3 Note that the (3, 3) entry of PCol(A) is unknown. ⇥ ⇤| (9 pts) (a) Is the system A| x = 3 3 0 0 consistent? ⇥ ⇤| (9 pts) (b) Find the projection of 9 0 0 onto Null(A| ). (7 pts) (c) Find the missing (3, 3) entry of PCol(A). Hint. Start by explaining the relationship between the the dimensions of Col(A| ) and Col(A). What property of projection matrices relates to dimension?  Problem 5. The following QR-factorization was calculated using the Gram-Schmidt algorithm. 2 A 3 2 Q 3 0 1 1 0 ⇤ 1/2 2p pR p 3 61 61/p2 2 2 2 2 6 0 57 7=6 p ⇤ 1/27 74 0 41 2 45 2 15 41/ 2 ⇤ 1/25 0 0 2 0 1 1 0 ⇤ 1/2 Note that the second column of Q is missing. (9 pts) (a) rank(A) = , rank(Q) = , and rank(R) = (8 pts) (b) Use the Gram-Schmidt algorithm to find the missing column of Q. You must use the Gram-Schmidt algorithm to receive any credit. ⇥ ⇤| (8 pts) (c) The vector b = 2 2 0 0 is orthogonal to the second column of Q. Find the least-squares approximate solution to Ax = b.  Problem 1. The equation below depicts the result of applying the Gauß-Jordan algorithm to the system represented by the augmented matrix [A | b].  x1 x2 x3 x4 x5  x1 x2 x3 x4 x5  →2 6 4 8 1 1 1 0 0 1 0 1  →8 3 8 3 →1 1  0 1 0 1 0 0      12 20 0 32 17 →5  1 rref  = 0 0 1 1 0   → 10 1 9 0 →4 3  →1    0 0 0 0 1   →6 →8 1 ↑ →7 2  0 0 0 0 0 0 3 3 →1 5 3 →1 0 0 0 0 0 0 Here, A represents the coe!cient matrix of the original system (so A is 6 ↓ 5). The vector b is the augmented column of the original system. Note that the (5, 4) entry of A is missing and marked as ↑. (4 pts) (a) Which sentence correctly describes the system Ax = b? ↔ inconsistent with no solutions ↔ consistent with exactly one solution ↔ consistent with infinitely many solutions ↔ inconsistent with infinitely many solutions (6 pts) (b) Fill in all of the blanks in the following figure. A R R rank (A ↭) = ( A) = r an k nullit = y (A ↭) y ( A) = nullit (4 pts) (c) The missing (5, 4) entry of A is ↑ =. (4 pts) (d ) Suppose we represent each row operation of this implementation of the Gauß-Jordan algorithm with elemen- tary matrices E1 , E2 ,... , Ek. Which of the following statements correctly describes E1→1 (the inverse of the elementary matrix representing the first elementary row operation in the algorithm)? ↔ 6 ↓ 6 representing →2 · r 1 ↗ r 1 ↔ 6 ↓ 6 representing →1/2 · r 1 ↗ r 1 ↔ 5 ↓ 5 representing →2 · r 1 ↗ r 1 ↔ 5 ↓ 5 representing →1/2 · r 1 ↗ r 1 ↔ 6 ↓ 6 representing r 1 + r 6 ↗ r 1 ' (↭ (8 pts) (e) Find all solutions x to Ax = b orthogonal to v = 1 1 1 1 1. (10 pts) Problem 2. Let A, S, and M be 2024 ↓ 2024 matrices where S is symmetric and M = (3 · I2024 → A)S + (4 · I2024 → SA)↭ A → A Suppose that v is an eigenvector of A corresponding to an eigenvalue ω = 1 and that this same vector v is in the null space of S. Show that v is also an eigenvector of M and identify the corresponding eigenvalue. Problem 3. The equation below depicts the calculation of A↭ A where A is the incidence matrix of a digraph G.  A↭    ↑ ↑ ↑ ↑  A  ↑ 2 1 →1 0 ↑ ↑ ↑ ↑ ↑ ↑  ↑ ↑ ↑↑  2 ↑ 1 →1 0 ↑ ↑ ↑ ↑ ↑    ↑ ↑ ↑  ↑ = 1 1 ↑ 1 →1 ↑  ↑ ↑ ↑ ↑  →1  ↑ ↑ ↑ →1 1 ↑ →1 ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 0 0 c →1 ↑ Note that all of the entries of A are unknown and marked as ↑. Also note that the diagonal entries of A↭ A are unknown and marked ast ↑ and that the (5, 3) entry of A↭ A is unknown and marked as c. (10 pts) (a) The Euler characteristic of G is , every diagonal entry of A↭ A is ↑ = , and c =. (5 pts) (b) Which figure below accurately depicts the way in which the first and last arrows a1 and a5 are configured in G? a1 ↔ ↔ a1 a5 a5 a1 ↔ a1 a5 ↔ a5 ↔ a1 a5  Problem 4. Consider the following EA = R factorization (note that these are all 5 ↓ 5 matrices).  E  A   R  ↑ ↑ ↑ ↑ ↑ 3 →12 18 14 ↑ 1 0 2 0 0 ↑ ↑ ↑ ↑ ↑  0 →3 3 3 ↑ 0 1 →1 0 c1       ↑ ↑ ↑ ↑ ↑  ↑  c2   →1 5 →7 →6  = 0 0 0 1  0 1 →3 3 →3→1 9 →11 →10 ↑ 0 0 0 0 0 0 1 0 0 1 0 3 →3 →3 ↑ 0 0 0 0 0 Note that several of the entries in E are unknown and marked as ↑. Also note that the last column of A is unknown and marked as ↑ and that two of the entries in R are unknown scalars labeled as c1 and c2. (4 pts) (a) rank(A) = (5 pts) (b) Select all of the nonpivot columns of A (no partial credit on this problem). ↔ column 1 ↔ column 2 ↔ column 3 ↔ column 4 ↔ column 5 (6 pts) (c) Only one of the following vectors makes the system Ax = b consistent. Select this vector.           1 0 0 0 1  0 3  1 0 1           ↔ b=  1  ↔ b= 0  ↔ b= →3  ↔ b= 1  ↔ b= 0   0 0  3 1 0 0 1 →3 0 0 (4 pts) (d ) The largest eigenvalue of R is ω =. ' (↭ (8 pts) (e) Calculate ↘v, w≃ where v = 0 1 0 0 1 and w is the fifth column of A (the one marked with ↑’s).  Problem 5. Consider the following P A = LU factorization (note that these are all 5 ↓ 5 matrices).  P    L  U  0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 0   1  0 0 0 1   1 0 0 0  0 2 2 2 0 0 0 0 1 0   1 1 1 0 0  0   A = 0 0 0 0  0 0 1 0 0    1  1 1 1 0  0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 (8 pts) (a) Select every matrix that has ω = 0 as an eigenvalue (2pts each). ↔ P ↔ A ↔ L ↔ U (4 pts) (b) trace(P →1 ) = (10 pts) (c) Determine whether or not A satisfies the equation A2024 → 5 A3 + 9 A → I5 = O Clearly explain your reasoning using ideas from class.  S y R Problem 1. This figure depicts displacement vectors v, w, x, and y between points P , Q, R, and S in R5 (so each of P , Q, R,