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Textbooks / Math / Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus 7

# Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus 7th Edition Solutions

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ISBN: 9781305253636

Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus | 7th Edition - Solutions by Chapter

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## Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus 7th Edition Student Assesment

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##### ISBN: 9781305253636

The full step-by-step solution to problem in Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus were answered by , our top Math solution expert on 10/03/18, 03:08PM. Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus was written by and is associated to the ISBN: 9781305253636. Since problems from 0 chapters in Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus have been answered, more than 200 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: Study Guide for Stewart/Redlin/Watson's Precalculus: Mathematics for Calculus, edition: 7. This expansive textbook survival guide covers the following chapters: 0.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Basis for V.

Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• 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

• Dimension of vector space

dim(V) = number of vectors in any basis for V.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Gram-Schmidt 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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• Least squares solution X.

The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

• Linear combination cv + d w or L C jV j.

• Lucas numbers

Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• 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.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• 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 p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Right inverse A+.

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

• 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!

• 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·

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.