 7.7.1: Simplify. See Example 1.281
 7.7.2: Simplify. See Example 1.249
 7.7.3: Simplify. See Example 1.27
 7.7.4: Simplify. See Example 1.23
 7.7.5: Simplify. See Example 1.  216
 7.7.6: Simplify. See Example 1. 24
 7.7.7: Simplify. See Example 1.264
 7.7.8: Simplify. See Example 1.2100
 7.7.9: Write in terms of i. See Example 1.224
 7.7.10: Write in terms of i. See Example 1.232
 7.7.11: Write in terms of i. See Example 1. 236
 7.7.12: Write in terms of i. See Example 1. 2121
 7.7.13: Write in terms of i. See Example 1.8263
 7.7.14: Write in terms of i. See Example 1.4220
 7.7.15: Write in terms of i. See Example 1. 254
 7.7.16: Write in terms of i. See Example 1.263
 7.7.17: Multiply or divide. See Example 2.22 # 27
 7.7.18: Multiply or divide. See Example 2.211 # 23
 7.7.19: Multiply or divide. See Example 2.25 # 210
 7.7.20: Multiply or divide. See Example 2.22 # 26
 7.7.21: Multiply or divide. See Example 2.216 # 21
 7.7.22: Multiply or divide. See Example 2.23 # 227
 7.7.23: Multiply or divide. See Example 2.2923
 7.7.24: Multiply or divide. See Example 2.249210
 7.7.25: Multiply or divide. See Example 2.280210
 7.7.26: Multiply or divide. See Example 2.24028
 7.7.27: Add or subtract. Write the sum or difference in the form a + bi. Se...
 7.7.28: Add or subtract. Write the sum or difference in the form a + bi. Se...
 7.7.29: Add or subtract. Write the sum or difference in the form a + bi. Se...
 7.7.30: Add or subtract. Write the sum or difference in the form a + bi. Se...
 7.7.31: Add or subtract. Write the sum or difference in the form a + bi. Se...
 7.7.32: Add or subtract. Write the sum or difference in the form a + bi. Se...
 7.7.33: Multiply. Write the product in the form a + bi. See Example 4.10i ...
 7.7.34: Multiply. Write the product in the form a + bi. See Example 4.2i #...
 7.7.35: Multiply. Write the product in the form a + bi. See Example 4.6i12 ...
 7.7.36: Multiply. Write the product in the form a + bi. See Example 4.5i14 ...
 7.7.37: Multiply. Write the product in the form a + bi. See Example 4. 23 +...
 7.7.38: Multiply. Write the product in the form a + bi. See Example 4.1 25 ...
 7.7.39: Multiply. Write the product in the form a + bi. See Example 4.14  ...
 7.7.40: Multiply. Write the product in the form a + bi. See Example 4.16  ...
 7.7.41: Write each quotient in the form a + bi. See Example 5.4i
 7.7.42: Write each quotient in the form a + bi. See Example 5.56i
 7.7.43: Write each quotient in the form a + bi. See Example 5.74 + 3i
 7.7.44: Write each quotient in the form a + bi. See Example 5.91  2i
 7.7.45: Write each quotient in the form a + bi. See Example 5.3 + 5i1 + i
 7.7.46: Write each quotient in the form a + bi. See Example 5.6 + 2i4  3
 7.7.47: Write each quotient in the form a + bi. See Example 5.5  i3  2i
 7.7.48: Write each quotient in the form a + bi. See Example 5.6  i2 + i
 7.7.49: Perform each indicated operation. Write the result in the form a + ...
 7.7.50: Perform each indicated operation. Write the result in the form a + ...
 7.7.51: Perform each indicated operation. Write the result in the form a + ...
 7.7.52: Perform each indicated operation. Write the result in the form a + ...
 7.7.53: Perform each indicated operation. Write the result in the form a + ...
 7.7.54: Perform each indicated operation. Write the result in the form a + ...
 7.7.55: Perform each indicated operation. Write the result in the form a + ...
 7.7.56: Perform each indicated operation. Write the result in the form a + ...
 7.7.57: Perform each indicated operation. Write the result in the form a + ...
 7.7.58: Perform each indicated operation. Write the result in the form a + ...
 7.7.59: Perform each indicated operation. Write the result in the form a + ...
 7.7.60: Perform each indicated operation. Write the result in the form a + ...
 7.7.61: Perform each indicated operation. Write the result in the form a + ...
 7.7.62: Perform each indicated operation. Write the result in the form a + ...
 7.7.63: Perform each indicated operation. Write the result in the form a + ...
 7.7.64: Perform each indicated operation. Write the result in the form a + ...
 7.7.65: Perform each indicated operation. Write the result in the form a + ...
 7.7.66: Perform each indicated operation. Write the result in the form a + ...
 7.7.67: Perform each indicated operation. Write the result in the form a + ...
 7.7.68: Perform each indicated operation. Write the result in the form a + ...
 7.7.69: Perform each indicated operation. Write the result in the form a + ...
 7.7.70: Perform each indicated operation. Write the result in the form a + ...
 7.7.71: Perform each indicated operation. Write the result in the form a + ...
 7.7.72: Perform each indicated operation. Write the result in the form a + ...
 7.7.73: Perform each indicated operation. Write the result in the form a + ...
 7.7.74: Perform each indicated operation. Write the result in the form a + ...
 7.7.75: Perform each indicated operation. Write the result in the form a + ...
 7.7.76: Perform each indicated operation. Write the result in the form a + ...
 7.7.77: Perform each indicated operation. Write the result in the form a + ...
 7.7.78: Perform each indicated operation. Write the result in the form a + ...
 7.7.79: Perform each indicated operation. Write the result in the form a + ...
 7.7.80: Perform each indicated operation. Write the result in the form a + ...
 7.7.81: Find each power of i. See Example 6.i 8
 7.7.82: Find each power of i. See Example 6.i 10
 7.7.83: Find each power of i. See Example 6.i 21
 7.7.84: Find each power of i. See Example 6.i 15
 7.7.85: Find each power of i. See Example 6.i 11
 7.7.86: Find each power of i. See Example 6.i 40
 7.7.87: Find each power of i. See Example 6.i 6
 7.7.88: Find each power of i. See Example 6.i 9
 7.7.89: Find each power of i. See Example 6.12i26
 7.7.90: Find each power of i. See Example 6.15i24
 7.7.91: Find each power of i. See Example 6.1 3i25
 7.7.92: Find each power of i. See Example 6. 1 2i27
 7.7.93: Recall that the sum of the measures of the angles of a triangle is ...
 7.7.94: Recall that the sum of the measures of the angles of a triangle is ...
 7.7.95: Use synthetic division to divide the following. See Section 6.4.1x3...
 7.7.96: Use synthetic division to divide the following. See Section 6.4.15x...
 7.7.97: Thirty people were recently polled about the average monthly balanc...
 7.7.98: Thirty people were recently polled about the average monthly balanc...
 7.7.99: Thirty people were recently polled about the average monthly balanc...
 7.7.100: Thirty people were recently polled about the average monthly balanc...
 7.7.101: Thirty people were recently polled about the average monthly balanc...
 7.7.102: Thirty people were recently polled about the average monthly balanc...
 7.7.103: Write in the form a + bi.i 3  i 4
 7.7.104: Write in the form a + bi.i 8  i 7
 7.7.105: Write in the form a + bi.i 6 + i 8
 7.7.106: Write in the form a + bi. i 4 + i 12
 7.7.107: Write in the form a + bi.2 + 29
 7.7.108: Write in the form a + bi.5  216
 7.7.109: Write in the form a + bi.6 + 2183
 7.7.110: Write in the form a + bi.4  282
 7.7.111: Write in the form a + bi.5  27510
 7.7.112: Write in the form a + bi.7 + 29814
 7.7.113: Describe how to find the conjugate of a complex number
 7.7.114: Explain why the product of a complex number and its complex conjuga...
 7.7.115: Simplify.18  232  12 + 2122
 7.7.116: 18  242  12 + 2162
 7.7.117: Determine whether 2i is a solution of x2 + 4 = 0
 7.7.118: Determine whether 1 + i is a solution of x2 + 2x = 2.
Solutions for Chapter 7.7: Complex Numbers
Full solutions for Intermediate Algebra  6th Edition
ISBN: 9780321785046
Solutions for Chapter 7.7: Complex Numbers
Get Full SolutionsIntermediate Algebra was written by and is associated to the ISBN: 9780321785046. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Since 118 problems in chapter 7.7: Complex Numbers have been answered, more than 55118 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.7: Complex Numbers includes 118 full stepbystep solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Outer product uv T
= column times row = rank one matrix.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).