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# Solutions for Chapter 8: College Algebra and Trigonometry 7th Edition

## Full solutions for College Algebra and Trigonometry | 7th Edition

ISBN: 9781439048603

Solutions for Chapter 8

Solutions for Chapter 8
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##### ISBN: 9781439048603

College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Since 71 problems in chapter 8 have been answered, more than 21193 students have viewed full step-by-step solutions from this chapter. Chapter 8 includes 71 full step-by-step solutions. This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Hilbert matrix hilb(n).

Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

• Hypercube matrix pl.

Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Incidence matrix of a directed graph.

The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

• 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.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• 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.

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Outer product uv T

= column times row = rank one matrix.

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• Pseudoinverse A+ (Moore-Penrose inverse).

The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

• Row picture of Ax = b.

Each equation gives a plane in Rn; the planes intersect at x.

• 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.

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).