- A.7.1: The square of a complex number is sometimes negative
- A.7.2: 12 + i2 12 - i2 =
- A.7.3: In the complex number system, a quadratic equation has four solutions.
- A.7.4: In the complex number 5 + 2i, the number 5 is called the part; the ...
- A.7.5: The equation x2 = -4 has the solution set
- A.7.6: The conjugate of 2 + 5i is -2 - 5i.
- A.7.7: All real numbers are complex numbers.
- A.7.8: If 2 - 3i is a solution of a quadratic equation with real coefficie...
- A.7.9: In 9 46, write each expression in the standard form a + bi.
- A.7.10: In 9 46, write each expression in the standard form a + bi.
- A.7.11: In 9 46, write each expression in the standard form a + bi.
- A.7.12: In 9 46, write each expression in the standard form a + bi.
- A.7.13: In 9 46, write each expression in the standard form a + bi.
- A.7.14: In 9 46, write each expression in the standard form a + bi.
- A.7.15: In 9 46, write each expression in the standard form a + bi.
- A.7.16: In 9 46, write each expression in the standard form a + bi.
- A.7.17: In 9 46, write each expression in the standard form a + bi.
- A.7.18: In 9 46, write each expression in the standard form a + bi.
- A.7.19: In 9 46, write each expression in the standard form a + bi.
- A.7.20: In 9 46, write each expression in the standard form a + bi.
- A.7.21: In 9 46, write each expression in the standard form a + bi.
- A.7.22: In 9 46, write each expression in the standard form a + bi.
- A.7.23: In 9 46, write each expression in the standard form a + bi.
- A.7.24: In 9 46, write each expression in the standard form a + bi.
- A.7.25: In 9 46, write each expression in the standard form a + bi.
- A.7.26: In 9 46, write each expression in the standard form a + bi.
- A.7.27: In 9 46, write each expression in the standard form a + bi.
- A.7.28: In 9 46, write each expression in the standard form a + bi.
- A.7.29: In 9 46, write each expression in the standard form a + bi.
- A.7.30: In 9 46, write each expression in the standard form a + bi.
- A.7.31: In 9 46, write each expression in the standard form a + bi.
- A.7.32: In 9 46, write each expression in the standard form a + bi.
- A.7.33: In 9 46, write each expression in the standard form a + bi.
- A.7.34: In 9 46, write each expression in the standard form a + bi.
- A.7.35: In 9 46, write each expression in the standard form a + bi.
- A.7.36: In 9 46, write each expression in the standard form a + bi.
- A.7.37: In 9 46, write each expression in the standard form a + bi.
- A.7.38: In 9 46, write each expression in the standard form a + bi.
- A.7.39: In 9 46, write each expression in the standard form a + bi.
- A.7.40: In 9 46, write each expression in the standard form a + bi.
- A.7.41: In 9 46, write each expression in the standard form a + bi.
- A.7.42: In 9 46, write each expression in the standard form a + bi.
- A.7.43: In 9 46, write each expression in the standard form a + bi.
- A.7.44: In 9 46, write each expression in the standard form a + bi.
- A.7.45: In 9 46, write each expression in the standard form a + bi.
- A.7.46: In 9 46, write each expression in the standard form a + bi.
- A.7.47: In 47 52, perform the indicated operations and express your answer ...
- A.7.48: In 47 52, perform the indicated operations and express your answer ...
- A.7.49: In 47 52, perform the indicated operations and express your answer ...
- A.7.50: In 47 52, perform the indicated operations and express your answer ...
- A.7.51: In 47 52, perform the indicated operations and express your answer ...
- A.7.52: In 47 52, perform the indicated operations and express your answer ...
- A.7.53: In 53 72, solve each equation in the complex number system.
- A.7.54: In 53 72, solve each equation in the complex number system.
- A.7.55: In 53 72, solve each equation in the complex number system.
- A.7.56: In 53 72, solve each equation in the complex number system.
- A.7.57: In 53 72, solve each equation in the complex number system.
- A.7.58: In 53 72, solve each equation in the complex number system.
- A.7.59: In 53 72, solve each equation in the complex number system.
- A.7.60: In 53 72, solve each equation in the complex number system.
- A.7.61: In 53 72, solve each equation in the complex number system.
- A.7.62: In 53 72, solve each equation in the complex number system.
- A.7.63: In 53 72, solve each equation in the complex number system.
- A.7.64: In 53 72, solve each equation in the complex number system.
- A.7.65: In 53 72, solve each equation in the complex number system.
- A.7.66: In 53 72, solve each equation in the complex number system.
- A.7.67: In 53 72, solve each equation in the complex number system.
- A.7.68: In 53 72, solve each equation in the complex number system.
- A.7.69: In 53 72, solve each equation in the complex number system.
- A.7.70: In 53 72, solve each equation in the complex number system.
- A.7.71: In 53 72, solve each equation in the complex number system.
- A.7.72: In 53 72, solve each equation in the complex number system.
- A.7.73: In 73 78, without solving, determine the character of the solutions...
- A.7.74: In 73 78, without solving, determine the character of the solutions...
- A.7.75: In 73 78, without solving, determine the character of the solutions...
- A.7.76: In 73 78, without solving, determine the character of the solutions...
- A.7.77: In 73 78, without solving, determine the character of the solutions...
- A.7.78: In 73 78, without solving, determine the character of the solutions...
- A.7.79: 2 + 3i is a solution of a quadratic equation with real coefficients...
- A.7.80: 4 - i is a solution of a quadratic equation with real coefficients....
- A.7.81: In 81 84, z = 3 - 4i and w = 8 + 3i. Write each expression in the s...
- A.7.82: In 81 84, z = 3 - 4i and w = 8 + 3i. Write each expression in the s...
- A.7.83: In 81 84, z = 3 - 4i and w = 8 + 3i. Write each expression in the s...
- A.7.84: In 81 84, z = 3 - 4i and w = 8 + 3i. Write each expression in the s...
- A.7.85: The impedance Z, in ohms, of a circuit element is defined as the ra...
- A.7.86: In an ac circuit with two parallel pathways, the total impedance Z,...
- A.7.87: Use z = a + bi to show that z + z = 2a and z - z = 2bi.
- A.7.88: Use z = a + bi to show that z = z
- A.7.89: Use z = a + bi and w = c + di to show that z + w = z + w
- A.7.90: Use z = a + bi and w = c + di to show that z # w = z # w.
- A.7.91: Explain to a friend how you would add two complex numbers and how y...
- A.7.92: Write a brief paragraph that compares the method used to rationaliz...
- A.7.93: Use an Internet search engine to investigate the origins of complex...
- A.7.94: A student multiplied 2-9 and 2-9 as follows:2-9 # 2-9 = 2(-9)(-9) =...
Solutions for Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System
Full solutions for Precalculus Enhanced with Graphing Utilities | 6th Edition
ISBN: 9780132854351
Solutions for Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System
Get Full Solutions
Solutions for Chapter A.7
16
1
Since 94 problems in chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System have been answered, more than 76709 students have viewed full step-by-step 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: Precalculus Enhanced with Graphing Utilities, edition: 6. Chapter A.7: Complex Numbers; Quadratic Equations in the Complex Number System includes 94 full step-by-step solutions. Precalculus Enhanced with Graphing Utilities was written by and is associated to the ISBN: 9780132854351.
Key Math Terms and definitions covered in this textbook
-
Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.
-
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
-
Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
-
Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
-
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.
-
Dimension of vector space
dim(V) = number of vectors in any basis for V.
-
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.
-
Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
-
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
-
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
-
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
-
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
-
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.
-
Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
-
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
-
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
-
Rotation matrix
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
-
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
-
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
-
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·