 1.5.1: Match the type of complex number with its definition.
 1.5.2: The imaginary unit is defined as ________, where ________.
 1.5.3: If is a positive number, the ________ ________ root of the negative...
 1.5.4: The numbers and are called ________ ________, and their product is ...
 1.5.5: In Exercises 5 8, find real numbers and such that the equation is true
 1.5.6: In Exercises 5 8, find real numbers and such that the equation is true
 1.5.7: In Exercises 5 8, find real numbers and such that the equation is true
 1.5.8: In Exercises 5 8, find real numbers and such that the equation is true
 1.5.9: In Exercises 920, write the complex number in standard form.
 1.5.10: In Exercises 920, write the complex number in standard form.
 1.5.11: In Exercises 920, write the complex number in standard form.
 1.5.12: In Exercises 920, write the complex number in standard form.
 1.5.13: In Exercises 920, write the complex number in standard form.
 1.5.14: In Exercises 920, write the complex number in standard form.
 1.5.15: In Exercises 920, write the complex number in standard form.
 1.5.16: In Exercises 920, write the complex number in standard form.
 1.5.17: In Exercises 920, write the complex number in standard form.
 1.5.18: In Exercises 920, write the complex number in standard form.
 1.5.19: In Exercises 920, write the complex number in standard form.
 1.5.20: In Exercises 920, write the complex number in standard form.
 1.5.21: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.22: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.23: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.24: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.25: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.26: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.27: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.28: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.29: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.30: In Exercises 2130, perform the addition or subtraction and write th...
 1.5.31: In Exercises 31 40, perform the operation and write the result in s...
 1.5.32: In Exercises 31 40, perform the operation and write the result in s...
 1.5.33: In Exercises 31 40, perform the operation and write the result in s...
 1.5.34: In Exercises 31 40, perform the operation and write the result in s...
 1.5.35: In Exercises 31 40, perform the operation and write the result in s...
 1.5.36: In Exercises 31 40, perform the operation and write the result in s...
 1.5.37: In Exercises 31 40, perform the operation and write the result in s...
 1.5.38: In Exercises 31 40, perform the operation and write the result in s...
 1.5.39: In Exercises 31 40, perform the operation and write the result in s...
 1.5.40: In Exercises 31 40, perform the operation and write the result in s...
 1.5.41: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.42: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.43: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.44: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.45: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.46: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.47: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.48: In Exercises 41 48, write the complex conjugate of the complex numb...
 1.5.49: In Exercises 4958, write the quotient in standard form.
 1.5.50: In Exercises 4958, write the quotient in standard form.
 1.5.51: In Exercises 4958, write the quotient in standard form.
 1.5.52: In Exercises 4958, write the quotient in standard form.
 1.5.53: In Exercises 4958, write the quotient in standard form.
 1.5.54: In Exercises 4958, write the quotient in standard form.
 1.5.55: In Exercises 4958, write the quotient in standard form.
 1.5.56: In Exercises 4958, write the quotient in standard form.
 1.5.57: In Exercises 4958, write the quotient in standard form.
 1.5.58: In Exercises 4958, write the quotient in standard form.
 1.5.59: In Exercises 5962, perform the operation and write the result in st...
 1.5.60: In Exercises 5962, perform the operation and write the result in st...
 1.5.61: In Exercises 5962, perform the operation and write the result in st...
 1.5.62: In Exercises 5962, perform the operation and write the result in st...
 1.5.63: In Exercises 6368, write the complex number in standard form.
 1.5.64: In Exercises 6368, write the complex number in standard form.
 1.5.65: In Exercises 6368, write the complex number in standard form.
 1.5.66: In Exercises 6368, write the complex number in standard form.
 1.5.67: In Exercises 6368, write the complex number in standard form.
 1.5.68: In Exercises 6368, write the complex number in standard form.
 1.5.69: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.70: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.71: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.72: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.73: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.74: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.75: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.76: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.77: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.78: In Exercises 6978, use the Quadratic Formula to solve the quadratic...
 1.5.79: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.80: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.81: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.82: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.83: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.84: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.85: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.86: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.87: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.88: In Exercises 7988, simplify the complex number and write it in stan...
 1.5.89: IMPEDANCE The opposition to current in an electrical circuit is cal...
 1.5.90: Cube each complex number. (a) 2 (b) (c)
 1.5.91: Raise each complex number to the fourth power. (a) 2 (b) (c) (d)
 1.5.92: Write each of the powers of as
 1.5.93: TRUE OR FALSE? In Exercises 9396, determine whether the statement i...
 1.5.94: TRUE OR FALSE? In Exercises 9396, determine whether the statement i...
 1.5.95: TRUE OR FALSE? In Exercises 9396, determine whether the statement i...
 1.5.96: TRUE OR FALSE? In Exercises 9396, determine whether the statement i...
 1.5.97: PATTERN RECOGNITION Complete the following. What pattern do you see...
 1.5.98: CAPSTONE Consider the binomials and and the complex numbers and (a)...
 1.5.99: ERROR ANALYSIS Describe the error.
 1.5.100: PROOF Prove that the complex conjugate of the product of two comple...
 1.5.101: PROOF Prove that the complex conjugate of the sum of two complex nu...
Solutions for Chapter 1.5: COMPLEX NUMBERS
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 1.5: COMPLEX NUMBERS
Get Full SolutionsSince 101 problems in chapter 1.5: COMPLEX NUMBERS have been answered, more than 32885 students have viewed full stepbystep solutions from this chapter. Chapter 1.5: COMPLEX NUMBERS includes 101 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 8. College Algebra was written by and is associated to the ISBN: 9781439048696.

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

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.