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# Solutions for Chapter 1-8: CUMULATIVE REVIEW EXERCISES

## Full solutions for Introductory & Intermediate Algebra for College Students | 4th Edition

ISBN: 9780321758941

Solutions for Chapter 1-8: CUMULATIVE REVIEW EXERCISES

Solutions for Chapter 1-8
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##### ISBN: 9780321758941

This 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. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Since 25 problems in chapter 1-8: CUMULATIVE REVIEW EXERCISES have been answered, more than 75014 students have viewed full step-by-step solutions from this chapter. Chapter 1-8: CUMULATIVE REVIEW EXERCISES includes 25 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• 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.

• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• Cholesky factorization

A = CTC = (L.J]))(L.J]))T for positive definite A.

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Diagonal matrix D.

dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

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

Use AT for complex A.

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Outer product uv T

= column times row = rank one matrix.

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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

Column vectors by convention.

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Subspace S of V.

Any vector space inside V, including V and Z = {zero vector only}.

• Volume of box.

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

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