 8.1.1: Concept Check Determine whether each statement is true or false. If...
 8.1.2: Concept Check Determine whether each statement is true or false. If...
 8.1.3: Concept Check Determine whether each statement is true or false. If...
 8.1.4: Concept Check Determine whether each statement is true or false. If...
 8.1.5: Concept Check Determine whether each statement is true or false. If...
 8.1.6: Concept Check Determine whether each statement is true or false. If...
 8.1.7: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.8: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.9: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.10: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.11: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.12: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.13: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.14: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.15: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.16: Identify each number as real, complex, pure imaginary, or nonreal c...
 8.1.17: Write each number as the product of a real number and i. See Exampl...
 8.1.18: Write each number as the product of a real number and i. See Exampl...
 8.1.19: Write each number as the product of a real number and i. See Exampl...
 8.1.20: Write each number as the product of a real number and i. See Exampl...
 8.1.21: Write each number as the product of a real number and i. See Exampl...
 8.1.22: Write each number as the product of a real number and i. See Exampl...
 8.1.23: Write each number as the product of a real number and i. See Exampl...
 8.1.24: Write each number as the product of a real number and i. See Exampl...
 8.1.25: Solve each quadratic equation and express all nonreal complex solut...
 8.1.26: Solve each quadratic equation and express all nonreal complex solut...
 8.1.27: Solve each quadratic equation and express all nonreal complex solut...
 8.1.28: Solve each quadratic equation and express all nonreal complex solut...
 8.1.29: Solve each quadratic equation and express all nonreal complex solut...
 8.1.30: Solve each quadratic equation and express all nonreal complex solut...
 8.1.31: Solve each quadratic equation and express all nonreal complex solut...
 8.1.32: Solve each quadratic equation and express all nonreal complex solut...
 8.1.33: Solve each quadratic equation and express all nonreal complex solut...
 8.1.34: Solve each quadratic equation and express all nonreal complex solut...
 8.1.35: Solve each quadratic equation and express all nonreal complex solut...
 8.1.36: Solve each quadratic equation and express all nonreal complex solut...
 8.1.37: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.38: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.39: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.40: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.41: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.42: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.43: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.44: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.45: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.46: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.47: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.48: Multiply or divide, as indicated. Simplify each answer. See Example...
 8.1.49: Write each number in standard form a + bi. See Example 5. 6  224 2
 8.1.50: Write each number in standard form a + bi. See Example 5. 9  218 3
 8.1.51: Write each number in standard form a + bi. See Example 5. 10 + 2200 5
 8.1.52: Write each number in standard form a + bi. See Example 5. 20 + 28 2
 8.1.53: Write each number in standard form a + bi. See Example 5. 3 + 218 24
 8.1.54: Write each number in standard form a + bi. See Example 5. 5 + 250 10
 8.1.55: Find each sum or difference. Write the answer in standard form. See...
 8.1.56: Find each sum or difference. Write the answer in standard form. See...
 8.1.57: Find each sum or difference. Write the answer in standard form. See...
 8.1.58: Find each sum or difference. Write the answer in standard form. See...
 8.1.59: Find each sum or difference. Write the answer in standard form. See...
 8.1.60: Find each sum or difference. Write the answer in standard form. See...
 8.1.61: Find each sum or difference. Write the answer in standard form. See...
 8.1.62: Find each sum or difference. Write the answer in standard form. See...
 8.1.63: Find each product. Write the answer in standard form. See Example 7...
 8.1.64: Find each product. Write the answer in standard form. See Example 7...
 8.1.65: Find each product. Write the answer in standard form. See Example 7...
 8.1.66: Find each product. Write the answer in standard form. See Example 7...
 8.1.67: Find each product. Write the answer in standard form. See Example 7...
 8.1.68: Find each product. Write the answer in standard form. See Example 7...
 8.1.69: Find each product. Write the answer in standard form. See Example 7...
 8.1.70: Find each product. Write the answer in standard form. See Example 7...
 8.1.71: Find each product. Write the answer in standard form. See Example 7...
 8.1.72: Find each product. Write the answer in standard form. See Example 7...
 8.1.73: Find each product. Write the answer in standard form. See Example 7...
 8.1.74: Find each product. Write the answer in standard form. See Example 7...
 8.1.75: Find each product. Write the answer in standard form. See Example 7...
 8.1.76: Find each product. Write the answer in standard form. See Example 7...
 8.1.77: Find each product. Write the answer in standard form. See Example 7...
 8.1.78: Find each product. Write the answer in standard form. See Example 7...
 8.1.79: Find each product. Write the answer in standard form. See Example 7...
 8.1.80: Find each product. Write the answer in standard form. See Example 7...
 8.1.81: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.82: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.83: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.84: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.85: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.86: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.87: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.88: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.89: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.90: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.91: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.92: Find each quotient. Write the answer in standard form a + bi. See E...
 8.1.93: Simplify each power of i. See Example 9. i 25
 8.1.94: Simplify each power of i. See Example 9. i 29
 8.1.95: Simplify each power of i. See Example 9. i 22
 8.1.96: Simplify each power of i. See Example 9. i 26
 8.1.97: Simplify each power of i. See Example 9. i 23
 8.1.98: Simplify each power of i. See Example 9. i 27
 8.1.99: Simplify each power of i. See Example 9. i 32
 8.1.100: Simplify each power of i. See Example 9. i 40
 8.1.101: Simplify each power of i. See Example 9. i 13
 8.1.102: Simplify each power of i. See Example 9. i 14
 8.1.103: Simplify each power of i. See Example 9. 1 i 11
 8.1.104: Simplify each power of i. See Example 9. 1 i 12
 8.1.105: Suppose that your friend, Kathy Strautz, tells you that she has dis...
 8.1.106: Explain why the following method of simplifying i 42 works. i 42 ...
 8.1.107: Show that 22 2 + 22 2 i is a square root of i.
 8.1.108: Show that 23 2 + 12 i is a cube root of i.
 8.1.109: Show that 2 + i is a solution of the equation x2 + 4x + 5 = 0.
 8.1.110: Show that 3 + 4i is a solution of the equation x2 + 6x + 25 = 0.
 8.1.111: (Modeling) Alternating Current Complex numbers are used to describe...
 8.1.112: (Modeling) Alternating Current Complex numbers are used to describe...
 8.1.113: (Modeling) Alternating Current Complex numbers are used to describe...
 8.1.114: (Modeling) Alternating Current Complex numbers are used to describe...
 8.1.115: The circuit contains two light bulbs and two electric motors. Assum...
 8.1.116: The phase angle u measures the phase difference between the voltage...
Solutions for Chapter 8.1: Complex Numbers
Full solutions for Trigonometry  10th Edition
ISBN: 9780321671776
Solutions for Chapter 8.1: Complex Numbers
Get Full SolutionsChapter 8.1: Complex Numbers includes 116 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 116 problems in chapter 8.1: Complex Numbers have been answered, more than 35672 students have viewed full stepbystep solutions from this chapter. Trigonometry was written by and is associated to the ISBN: 9780321671776. This textbook survival guide was created for the textbook: Trigonometry, edition: 10.

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.

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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.