Additional Homework Problems

Transcription

1 Math Fall Additional Homework Problems 1 In parts (a) and (b) assume that the given system is consistent For each system determine all possibilities for the numbers r and n r where r is the number of nonzero rows of the (reduced) row echelon form of the augmented matrix and n is the number of the unknowns of the given system What do the values of r and n r have to do with the numbers of pivot variables and free variables in these systems? (a) Ax = b where A = [a ij ] 3 2 (b) Ax = b where A = [a ij ] In each of the following write the given expression in the form a + ib: (a) 3+i 4 5i, (b) exp(2 3i), (c) eiπ, (d) e 2 iπ/2, (e) 2 1 i, (f) π 1+2i 3 Solve each of the following equations: (a) x 4 + 2x = 0, (b) x = 0 4 Your friend Bob remembers a theorem from class about solutions to a homogeneous linear differential equation being linearly independent if and only if their Wronskian is always nonzero But he also checks for himself that x and x 2 are solutions to the same homogeneous linear differential equation (x 2 y 2xy + 2y = 0), and are linearly independent but their Wronskian is w(x) = x 2, which is NOT always nonzero Bob feels like he has found a counterexample to the theorem Explain why Bob is wrong 5 Your friend Bob is again thinking about the theorem from class about solutions to a homogeneous linear differential equation being linearly independent if and only if their Wronskian is always nonzero This time he notes that x and x 2 are solutions to the same homogeneous linear differential equation (y + x 2 y 2xy + 2y = 0), and are linearly independent but their Wronskian is w(x) = x 2, which is NOT always nonzero Bob again feels like he has found a counterexample to the theorem Explain why Bob is again wrong

2 6 For this problem, we consider the linear transformation T : R 2 R 2 defined by ( ) ( ) x 11x 18y T = y 3x 4y Let S be the standard basis for R 2, and let v 1 = of the vectors { v 1, v 2 } ( ) 2 and v 1 2 = ( ) 3 Let V be the basis for R 1 2 consisting (These are called eigenval- (a) Find the values λ 1 and λ 2 for which T ( v 1 ) = λ 1 v 1 and T ( v 2 ) = λ 2 v 2 ues and eigenvectors of the linear transformation T ) (b) Compute A = [T ] S S, [ v 1] S, and [ v 2 ] S (c) Write the two equations in part (a), with respect to the standard basis S (as matrix times vector equals scalar times vector) (d) Compute D = [T ] V V, [ v 1] V, and [ v 2 ] V (e) Write the two equations in part (a), with respect to the basis V (as matrix times vector equals scalar times vector) (f) Compute [I] S V and [I]V S (g) The matrices A and D are similar, by a conjugation representing the corresponding change of basis We could write this conjugation the following four forms: Compute P, Q, R, and S A = P 1 DP D = Q 1 AQ A = RDR 1 D = SAS 1 (h) What are the eigenvectors of A, and how are they represented in the matrix Q? (i) What are the eigenvalues of A, and how are they represented in the matrix D? (j) What are the eigenvectors and eigenvalues of D?

3 7 The matrix A represents the linear transformation T : R 6 R 6 with respect to the standard basis, and the Jordan form matrix J below represents the same linear transformation T with respect to the basis V = { v 1, v 2, v 3, v 4, v 5, v 6 } J = [T ] V V = (a) Identify the eigenvalue blocks and basic Jordan blocks in this Jordan form matrix J (b) Use the matrix J to write the images of the V vectors, T ( v 1 ),, T ( v 6 ), as linear combinations of the V vectors (c) Consider now the new basis V 1 = { v 5, v 6, v 1, v 2, v 3, v 4 }, consisting of the same vectors as V but reordered as indicated Use the information from (b) to compute J 1 = [T ] V 1 V 1 (d) Is J 1 a Jordan form matrix? Identify the eigenvalue blocks and basic Jordan blocks in J 1 (Note that these are the same blocks as in J, just rearranged) (e) Consider the Jordan form matrix J 2 below Use a reordering of V to find a basis V 2 such that J 2 = [T ] V J 2 = V 2 (f) Consider now the new basis V 3 = { v 6, v 5, v 1, v 2, v 3, v 4 }, with the indicated order Compute J 3 = [T ] V 3 V 3 Is this a Jordan form matrix? (g) The reorderings of the vectors to form the bases V 1 and V 2 show that a given matrix A can have several Jordan canonical forms, such as in this case the matrices J 1 and J 2 ; those basis reorderings evidently caused rearrangements of the eigenvalue blocks and basic Jordan blocks Any of these sorts of rearrangements of these blocks can be achieved by corresponding reorderings of the basis; specifically, eigenvalue blocks can be rearranged, and basic Jordan blocks inside of eigenvalue blocks can be rearranged J 3 however illustrates though that not all basis reorderings yield Jordan form matrices The Jordan canonical form of a matrix A is thus clearly not unique; but it is unique up to rearrangements of the sorts illustrated by the matrices J 1 and J 2 That is, any two Jordan forms similar to the matrix A are the same by way of rearranging the eigenvalue blocks, and/or rearranging the basic Jordan blocks inside of eigenvalue blocks, by way of a reordering of the basis

4 With this fact in mind, consider the matrix M given by M = Is this a Jordan form matrix? Identify the eigenvalue blocks and the basic Jordan blocks Is this matrix M similar to A? (h) Suppose that the vectors v 1, v 2, v 3, v 4, v 5, v 6, written with respect to the standard basis, are the columns of the matrix R below Use a conjugation expressing the similarity of A and J to compute the matrix A (You are encouraged to use matlab for this calculation) R = (i) Use a conjugation expressing the similarity of A and J 1 to compute the matrix A (You are encouraged to use matlab for this calculation) 8 The characteristic polynomial for the matrix A tells you (from its factorization) that the Jordan form has a 3x3 eigenvalue block with eigenvalue 4, a 5x5 eigenvalue block with eigenvalue 6, and a 2x2 eigenvalue block with eigenvalue 1 Without knowing anything more, how many different (non-similar) possible Jordan forms are there? (Keep in mind that reorderings/rearrangements from AHP 7 that make many Jordan form matrices similar should be viewed as the same Jordan form, in this sense) 9 Considering the matrix from problem 8, suppose that you find by calculation that the eigenspace E 6 has dimension 2 How many remaining possibilities are there for the Jordan form? 10 Find an easy example of nonzero matrices A, B, and C such that AB = AC, but B C 11 Find an easy example of nonzero matrices D, E, and F such that ED = F D, but E F 12 A matrix A acts on a vector x by multiplication to result in A x What would it mean to say that this action is linear? Is it true that this action is linear? Prove or find a counterexample 13 The plane P has equation n x = 0, and n is a unit vector The orthogonal projection of a vector

5 v to this plane is given by v ( v n) n What would it mean to say that vectors project linearly? Do vectors in fact project linearly? Prove or find a counterexample ( ) v 14 The normalization of a vector v results in the unit vector What would it mean to say that v normalization is linear? Is normalization actually linear? Prove or find a counterexample 15 Suppose that zero is an eigenvalue of the matrix A What does that tell you about the nullspace of A? What can you conclude from this about the question of whether A is or is not invertible? 16 Suppose that A is invertible What does this tell you about the nullspace of A? What can you conclude from this about the question of whether zero is or is not an eigenvalue of A? 17 The matrix A has Jordan form F below, obtained by way of the Jordan basis β below F = and β = 1 2, 1, 3, Find the matrix A, and a fundamental set of solutions to the system y = A y (you may use Matlab to help you with the arithmetic) 18 The matrix A below can be put into Jordan form by way of the Jordan basis β below A = and β = 1 2, 1, 3, Find a fundamental set of solutions to the system y = A y (you may use Matlab to help you with the arithmetic) 19 The arithmetic below is given Use this information to find a fundamental set of solutions to the system y = A y A = = =