 9.2.1: Find e2ni.
 9.2.2: Find<?(1/2)7r/.
 9.2.3: Write z = 1 + i in polar form as z = re,e.
 9.2.4: Sketch the trajectory of the complexvalued functionz = e3//.What i...
 9.2.5: Sketch the trajectory of the complexvalued functionz = ,(0.12/), ^
 9.2.6: Find all complex solutions of the systemdx It3 2 5 3in the form g...
 9.2.7: Determine the stability of the systemdx_ dt1 2 3 4
 9.2.8: Consider a systemdx a = Ax, dtx.where A is a symmetric matrix. When...
 9.2.9: Consider a systemdx = Ax. dtwhere A is a 2 x 2 matrix with tr A < 0...
 9.2.10: Consider a quadratic form q(x) = x Ax of two variables, jc i and *2...
 9.2.11: Do parts (a) and (d) of Exercise 10 for a quadratic form of n varia...
 9.2.12: Determine the stability of the systemdx ~dt0 1 0 0 0 1 1 1 2
 9.2.13: If the system dx/dt = Ax is stable, is dx/dt = A stable as well? Ho...
 9.2.14: Negative Feedback Loops. Suppose some quantities *1 (0 , x2(t), .....
 9.2.15: Consider a noninvertible 2 x 2 matrix A with a positive trace. What...
 9.2.16: Consider the systemwhere a and b are arbitrary constants. For which...
 9.2.17: Consider the systemdx ~dt1 k k 1where k is an arbitrary constant....
 9.2.18: Consider a diagonalizable 3 x 3 matrix A such that the zero state i...
 9.2.19: True or False ? If the trace and the determinant of a 3 x 3 matrix ...
 9.2.20: Consider a 2 x 2 matrix A with eigenvalues 7xi. Let v + iw be an ei...
 9.2.21: Ngozi opens a bank account w i t h an initial balance of 1,000 Nige...
 9.2.22: For each of the linear systems in Exercises 22 through 26, find the...
 9.2.23: For each of the linear systems in Exercises 22 through 26, find the...
 9.2.24: For each of the linear systems in Exercises 22 through 26, find the...
 9.2.25: For each of the linear systems in Exercises 22 through 26, find the...
 9.2.26: For each of the linear systems in Exercises 22 through 26, find the...
 9.2.27: Find all real solutions of the systems in Exercises 27 through 30.
 9.2.28: Find all real solutions of the systems in Exercises 27 through 30.
 9.2.29: Find all real solutions of the systems in Exercises 27 through 30.
 9.2.30: Find all real solutions of the systems in Exercises 27 through 30.
 9.2.31: Solve the systems in Exercises 31 through 34. Give the solution in ...
 9.2.32: Solve the systems in Exercises 31 through 34. Give the solution in ...
 9.2.33: Solve the systems in Exercises 31 through 34. Give the solution in ...
 9.2.34: Solve the systems in Exercises 31 through 34. Give the solution in ...
 9.2.35: Prove the product rule for derivatives of complexvalued functions.
 9.2.36: Consider the following massspring system:Let .r(r) be the deviatio...
 9.2.37: a. For a differentiable complexvalued function z(t)y find the deri...
 9.2.38: Let z\(t) and zi(t) be two complexvalued solutions of the initial ...
 9.2.39: Solve the systemdx ~dtCompare this with Exercise 9.1.24. When is th...
 9.2.40: An eccentric mathematician is able to gain autocratic power in a sm...
Solutions for Chapter 9.2: The Complex Case: Eulers Formula
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 9.2: The Complex Case: Eulers Formula
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since 40 problems in chapter 9.2: The Complex Case: Eulers Formula have been answered, more than 14944 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Chapter 9.2: The Complex Case: Eulers Formula includes 40 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.