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# Solutions for Chapter 2-8: Solving for a Specific Variable

## Full solutions for Algebra 1, Student Edition (MERRILL ALGEBRA 1) | 1st Edition

ISBN: 9780078738227

Solutions for Chapter 2-8: Solving for a Specific Variable

Solutions for Chapter 2-8
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##### ISBN: 9780078738227

Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. Since 48 problems in chapter 2-8: Solving for a Specific Variable have been answered, more than 35570 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2-8: Solving for a Specific Variable includes 48 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Diagonal matrix D.

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

• 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.

• 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.

• 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 .

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• 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.

• Orthogonal subspaces.

Every v in V is orthogonal to every w in W.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• Positive definite matrix A.

Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

P = aaT laTa has rank l.

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Column vectors by convention.

• 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.

• Singular Value Decomposition

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

• 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·

• Wavelets Wjk(t).

Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).

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