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 10.7.10.1.837: In Exercises 116, express each number in terms of i and simplify, i...
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 10.7.10.1.869: In Exercises 3362, find each product. Write imaginary results in th...
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 10.7.10.1.899: In Exercises 6384, divide and simplify to the form a bi. 2 3 i
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 10.7.10.1.910: In Exercises 6384, divide and simplify to the form a bi. 2 3i 3 i
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 10.7.10.1.912: In Exercises 6384, divide and simplify to the form a bi. 6 3i 4 2i
 10.7.10.1.913: In Exercises 6384, divide and simplify to the form a bi. 4 5i 3 7i
 10.7.10.1.914: In Exercises 6384, divide and simplify to the form a bi. 5 i 3 2i
 10.7.10.1.915: In Exercises 6384, divide and simplify to the form a bi. 7 3i
 10.7.10.1.916: In Exercises 6384, divide and simplify to the form a bi. 5 7i
 10.7.10.1.917: In Exercises 6384, divide and simplify to the form a bi. 8 5i 2i
 10.7.10.1.918: In Exercises 6384, divide and simplify to the form a bi. 3 4i 5i
 10.7.10.1.919: In Exercises 6384, divide and simplify to the form a bi. 4 7i 3i
 10.7.10.1.920: In Exercises 6384, divide and simplify to the form a bi. 5 i 4i
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 10.7.10.1.945: Let f (x) x2 2x 2. Find f (1 i).
 10.7.10.1.946: Let f (x) x2 2x 5. Find f (1 2i). a bi.
 10.7.10.1.947: In Exercises 111114, simplify each evaluation to the form a bi. Let...
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 10.7.10.1.951: Complex numbers are used in electronics to describe the current in ...
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 10.7.10.1.953: The mathematician Girolamo Cardano is credited with the first use (...
 10.7.10.1.954: What is the imaginary unit i?
 10.7.10.1.955: Explain how to write 64 as a multiple of i.
 10.7.10.1.956: What is a complex number? Explain when a complex number is a real n...
 10.7.10.1.957: Explain how to add complex numbers. Give an example.
 10.7.10.1.958: Explain how to subtract complex numbers. Give an example.
 10.7.10.1.959: Explain how to find the product of 2i and 5 3i.
 10.7.10.1.960: Explain how to find the product of 3 2i and 5 3i.
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 10.7.10.1.966: A standup comedian uses algebra in some jokes, including one about...
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 10.7.10.1.980: Simplify: x y2 1 y y x2 1 x . (Section 7.5, Example 3 or Example 6)
 10.7.10.1.981: Solve for x: 1 x 1 y 1 z . (Section 7.6, Example 7)
 10.7.10.1.982: If f(x) 2x2 x and g(x) x 6, find (g f)(3). (Section 8.3, Example 4)
 10.7.10.1.983: Exercises 147149 will help you prepare for the material covered in ...
 10.7.10.1.984: Exercises 147149 will help you prepare for the material covered in ...
 10.7.10.1.985: Exercises 147149 will help you prepare for the material covered in ...
Solutions for Chapter 10.7: Complex Numbers
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 10.7: Complex Numbers
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Chapter 10.7: Complex Numbers includes 160 full stepbystep solutions. Since 160 problems in chapter 10.7: Complex Numbers have been answered, more than 68316 students have viewed full stepbystep solutions from this chapter. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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].

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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 •

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.

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!

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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