 5.4.1: Simplify.
 5.4.2: Simplify.
 5.4.3: Simplify.
 5.4.4: Simplify.
 5.4.5: Simplify.
 5.4.6: Simplify.
 5.4.7: Simplify.
 5.4.8: Simplify.
 5.4.9: Simplify.
 5.4.10: Simplify.
 5.4.11: Solve each equation.
 5.4.12: Solve each equation.
 5.4.13: Find the values of m and n that make each equation true.
 5.4.14: Find the values of m and n that make each equation true.
 5.4.15: The current in one part of a series circuit is 4  j amps. The curr...
 5.4.16: Simplify.
 5.4.17: Simplify.
 5.4.18: Simplify.
 5.4.19: Simplify.
 5.4.20: Simplify.
 5.4.21: Simplify.
 5.4.22: Simplify.
 5.4.23: Simplify.
 5.4.24: Simplify.
 5.4.25: Simplify.
 5.4.26: Simplify.
 5.4.27: Simplify.
 5.4.28: Simplify.
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 5.4.30: Simplify.
 5.4.31: Simplify.
 5.4.32: Simplify.
 5.4.33: Simplify.
 5.4.34: Simplify.
 5.4.35: Simplify.
 5.4.36: Simplify.
 5.4.37: Simplify.
 5.4.38: Simplify.
 5.4.39: Simplify.
 5.4.40: Simplify.
 5.4.41: Simplify.
 5.4.42: Solve each equation.
 5.4.43: Solve each equation.
 5.4.44: Solve each equation.
 5.4.45: Solve each equation.
 5.4.46: Find the values of m and n that make each equation true.
 5.4.47: Find the values of m and n that make each equation true.
 5.4.48: Find the values of m and n that make each equation true.
 5.4.49: Find the values of m and n that make each equation true.
 5.4.50: The current in a circuit is 2 + 5j amps, and the impedance is 4  j...
 5.4.51: The voltage in a circuit is 14  8j volts, and the impedance is 2 ...
 5.4.52: Find the sum of i x 2  (2 + 3i)x + 2 and 4 x 2 + (5 + 2i)x  4i
 5.4.53: Simplify [(3 + i) x 2  ix + 4 + i]  [(2 + 3i) x 2 + (1  2i)x  3]
 5.4.54: Simplify
 5.4.55: Simplify
 5.4.56: Simplify
 5.4.57: Simplify
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 5.4.62: Simplify
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 5.4.64: Simplify
 5.4.65: Simplify
 5.4.66: Solve each equation, and locate the complex solutions in the comple...
 5.4.67: Solve each equation, and locate the complex solutions in the comple...
 5.4.68: Solve each equation, and locate the complex solutions in the comple...
 5.4.69: Solve each equation, and locate the complex solutions in the comple...
 5.4.70: Find the values of m and n that make each equation true.
 5.4.71: Find the values of m and n that make each equation true.
 5.4.72: The impedance in one part of a series circuit is 3 + 4j ohms, and t...
 5.4.73: Write two complex numbers with a product of 10.
 5.4.74: Copy and complete the table. Power of i Simplified Expression i 6 ?...
 5.4.75: Identify the expression that does not belong with the other three. ...
 5.4.76: Determine if each statement is true or false. If false, find a coun...
 5.4.77: Use the information on page 261 to explain how complex numbers are ...
 5.4.78: The area of the square is 16 square units. What is the area of the ...
 5.4.79: If i 2 = 1, then what is the value of i 71 ? F 1 G 0 H i J i
 5.4.80: Write a quadratic equation with the given root(s). Write the equati...
 5.4.81: Write a quadratic equation with the given root(s). Write the equati...
 5.4.82: Solve each equation by graphing. If exact roots cannot be found, st...
 5.4.83: Solve each equation by graphing. If exact roots cannot be found, st...
 5.4.84: Triangle ABC is reflected over the xaxis
 5.4.85: Triangle ABC is reflected over the xaxis
 5.4.86: Triangle ABC is reflected over the xaxis
 5.4.87: Triangle ABC is reflected over the xaxis
 5.4.88: A new sofa, love seat, and coffee table cost $2050. The sofa costs ...
 5.4.89: Samantha is going to use more than 75 but less than 100 bricks to m...
 5.4.90: Samantha is going to use more than 75 but less than 100 bricks to m...
 5.4.91: Determine whether each polynomial is a perfect square trinomial
 5.4.92: Determine whether each polynomial is a perfect square trinomial
 5.4.93: Determine whether each polynomial is a perfect square trinomial
 5.4.94: Determine whether each polynomial is a perfect square trinomial
 5.4.95: Determine whether each polynomial is a perfect square trinomial
Solutions for Chapter 5.4: Complex Numbers
Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving  1st Edition
ISBN: 9780078778568
Solutions for Chapter 5.4: Complex Numbers
Get Full SolutionsSince 95 problems in chapter 5.4: Complex Numbers have been answered, more than 41964 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568. Chapter 5.4: Complex Numbers includes 95 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).