MATH 308 C: Homework
For an \( (n \times n) \) matrix \( A \) the exponential of \(A\) is defined as \(e^A = \sum_{k = 0}^{\infty} \frac{A^k}{k!}\). If \(D = \begin{bmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & 0 & \cdots \\ \vdots & 0 & \ddots & 0 \\ 0 & \cdots & 0 & d_n \end{bmatrix} \), then \(e^D = \begin{bmatrix} e^{d_1} & 0 & \cdots & 0 \\ 0 & e^{d_2} & 0 & \cdots \\ \vdots & 0 & \ddots & 0 \\ 0 & \cdots & 0 & e^{d_n} \end{bmatrix} \). Let \(A\) be a diagonalizable matrix, that is, \(A= SDS^{-1} \) where \(D\) is a diagonal matrix. Then \(e^A = S e^D S^{-1} \). Compute \(e^A \) for problems 5 and 7 of 4.7.
Homework 8: Not Due § 4.4: 14 § 4.5: 6, 10, 14 (ignore the part of defective), 20, 22, § 4.7: 5, 7, 25
Note the due date. This is why the HW is longer than usual.
Homework 7: Nov 30 § 3.8: 1, 3 § 3.9: 1, 5, 11, 14 § 4.1: 2, 10, 14, 17 § 4.2: 12, 18, 24, 25, 28, 30 § 4.3: 4, 12
Homework 6: Nov 16 § 3.6: 4, 6, 10, 16, 22, 28 § 3.7: 3, 10, 14, 22, 28, 30, 36, 44 Also do: Let \(P_2\) be the vector space of polynomials of degree less than or equal to \(2\). Show that the function \(T: P_2 \to \mathbb{R}^2 \) given by \(T(p(x)) = \begin{bmatrix} \int^1_0 p(x) dx \\ p(0)\end{bmatrix}\) is a linear transformation.
Homework 5: Nov 9 § 3.4: 1, 9c, 11, 24, 34, 37 § 3.5: 4, 6, 9, 18, 26, 32, 33, 34
Comment on HW 4: For the problems on section 1.9 you CAN use any of the equivalent definitions for nonsingular/invertible, use the one that helps you the most to solve the problem.
Homework 4: Oct 26 § 1.9: 4, 8, 10, 15, 19, 20, 38, 39, 55, 70, 76 § 3.1: 13, 16, 21, 22 § 3.3: 19, 20, NOT DUE:(25, 32, 43, 48, 51, 52)
Comment on HW 3: When you do the problems on section 1.7, DO NOT use that nonsingular = invertible yet, as we don't see that until section 1.9.
Homework 3: Oct 19 § 1.7: 12, 14, 20, 23, 50, 56
Homework 2: Oct 12 § 1.5: 6, 8c, 10c, 12b, 20, 24, 30, 48 § 1.6: 3, 11, 26, 27, 40, 47
Hints for HW: § 1.2 #49: Recall that we can write any 3-digit number as the sum \(100*x+10*y+z\) where \(x,y\) and \(z\) are the digits. For example \(308\) can be written as \(100*3+10*0+8\) in this case \(x =3, y = 0\) and \(z = 8\). § 1.2 #51: Let \(x_1, x_2, x_3\) be the initial amount of money players one, two and three respectively had. Also assume that player one losses the first round, player two the second round and player three the last round. Then subtract or add the amount of money each player would have in terms of the unknowns in each step. In the last step the amount of money each of the have would equal \(24\). These relations yield a linear system.
Homework 1: Oct 5 § 1.1: 17, 29, 32, 35 § 1.2: 10, 21, 31, 35, 39, 49, 51 § 1.3: 4, 12, 18, 23, 24