Homework Answers
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find...
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation ) Ax 1 = and find the least squares approximation x' with minimum norm
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation...
How can I get the (a) 3*2 matrix A? x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0...
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
1 point) Read 'Diagonalization Changing to a Basis of Eigenvectors' before attempting this problem. Suppose that V is a 5-dimensional vector space. Let S -(vi,... , vs) be some ordered basis...
1 point) Read 'Diagonalization Changing to a Basis of Eigenvectors' before attempting this problem. Suppose that V is a 5-dimensional vector space. Let S -(vi,... , vs) be some ordered basis of V, and let T-(wi.... . ws) be some other ordered basis of V. Let L: V → V be a linear transformation. Let M be the matrix of L in the basis Sand et N be the matrix of L in the basis T. Decide whether each of...
nsid r the following et ār vnctors. Let 1 v2 and V3 be column vectors in and let A be the 3 × 3 matrix v 1 v2 v③ with these vectors as its columns. The vi v2 and ] are linearly dependent if and nly t...
nsid r the following et ār vnctors. Let 1 v2 and V3 be column vectors in and let A be the 3 × 3 matrix v 1 v2 v③ with these vectors as its columns. The vi v2 and ] are linearly dependent if and nly the hom 9ene us linear system with augmented matrix 시 has a no tr ia solution Consider the following equation. 81-3-311 Solve for ci 2, andc3. If a nontrlvial solution exists, state it or...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pse...
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pseudo-inverse of a vector u R" regarded as an m x 1 matrix? 3. (5 pts) Let B AT where A is the matrix in Problem 1. Use Matlab to find the singular value decomposition and the Moore-Penrose pseudo-inverse of B. Then solve minimum-norm least squares problem minl-ll : FE R minimizes IBr-ey where c- [1,2. Compare...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and correspondin...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, where...
Problem 1 part II and Problem 2 part I and II Problem 1: (Short Answer) 6 pts] The region R is bounded by y 0, 0,...
Problem 1 part II and Problem 2 part I and II
Problem 1: (Short Answer) 6 pts] The region R is bounded by y 0, 0, 2, and y- 2 4 1, 3 pts] If R is revolved about the line x = 5. If an integral or sunn of integrals with respect to z is used to calculate the volume, explain whether the washer or shell method should be used II. 3 pts) Suppose that R is the base...
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation ) Ax 1 = and find the least squares approximation x' with minimum norm
1 -1 Let A= -1 -2 1 1 singular value decomposition A = U£VT (a) Find a (b) Determine the pseudoinverse matrix At, expressing At single matrix. as a (c) Consider the equation...
How can I get the (a) 3*2 matrix A?
x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
1 point) Read 'Diagonalization Changing to a Basis of Eigenvectors' before attempting this problem. Suppose that V is a 5-dimensional vector space. Let S -(vi,... , vs) be some ordered basis of V, and let T-(wi.... . ws) be some other ordered basis of V. Let L: V → V be a linear transformation. Let M be the matrix of L in the basis Sand et N be the matrix of L in the basis T. Decide whether each of...
nsid r the following et ār vnctors. Let 1 v2 and V3 be column vectors in and let A be the 3 × 3 matrix v 1 v2 v③ with these vectors as its columns. The vi v2 and ] are linearly dependent if and nly the hom 9ene us linear system with augmented matrix 시 has a no tr ia solution Consider the following equation. 81-3-311 Solve for ci 2, andc3. If a nontrlvial solution exists, state it or...
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
4. For this question, we define the following matrices: 1-2 0 To 61 C= 0 -1 2 , D= 3 1 . [3 24 L-2 -1] (a) For each of the following, state whether or not the expression can be evaluated. If it can be, evaluate it. If it cannot be, explain why. i. B? +D ii. AD iii. C + DB iv. CT-C (b) Find three distinct vectors X1, X2, X3 such that Bx; = 0 for i =...
Let A e Rmxn. The linear system Ax = b can have either: (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions. If A is square and invertible, there is a unique solution, which can be written as x = A-'b. The concept of pseudoinverse seeks to generalise this idea to non-square matrices and to cases (ii) and (iii). Taking case (ii) of an inconsistent linear system, we may solve the normal equations AT Ar = Ab...
2. (5 pts) Assume A E Rm** with m > n has (full) rank n. Show that At = (ATA)TAT, What is the pseudo-inverse of a vector u R" regarded as an m x 1 matrix? 3. (5 pts) Let B AT where A is the matrix in Problem 1. Use Matlab to find the singular value decomposition and the Moore-Penrose pseudo-inverse of B. Then solve minimum-norm least squares problem minl-ll : FE R minimizes IBr-ey where c- [1,2. Compare...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, where...
Problem 1 part II and Problem 2 part I and II
Problem 1: (Short Answer) 6 pts] The region R is bounded by y 0, 0, 2, and y- 2 4 1, 3 pts] If R is revolved about the line x = 5. If an integral or sunn of integrals with respect to z is used to calculate the volume, explain whether the washer or shell method should be used II. 3 pts) Suppose that R is the base...
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