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Solutions for Chapter 3.6: Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions

Full solutions for Intermediate Algebra | 6th Edition

ISBN: 9780321785046

Solutions for Chapter 3.6: Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions

Solutions for Chapter 3.6
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ISBN: 9780321785046

Since 70 problems in chapter 3.6: Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions have been answered, more than 66759 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.6: Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions includes 70 full step-by-step solutions. This textbook survival guide was created for the textbook: Intermediate Algebra, edition: 6. Intermediate Algebra was written by and is associated to the ISBN: 9780321785046.

Key Math Terms and definitions covered in this textbook
• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Column space C (A) =

space of all combinations of the columns of A.

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• Fourier matrix F.

Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• 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.

• 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.

• 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).

• Normal equation AT Ax = ATb.

Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• 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!

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).