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Solutions for Chapter 4-1: Steepness of a Line

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

ISBN: 9780078738227

Solutions for Chapter 4-1: Steepness of a Line

Solutions for Chapter 4-1
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ISBN: 9780078738227

Algebra 1, Student Edition (MERRILL ALGEBRA 1) was written by and is associated to the ISBN: 9780078738227. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Algebra 1, Student Edition (MERRILL ALGEBRA 1) , edition: 1. Chapter 4-1: Steepness of a Line includes 76 full step-by-step solutions. Since 76 problems in chapter 4-1: Steepness of a Line have been answered, more than 33957 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

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

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• 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.

• 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.

• Hankel matrix H.

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

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• 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.

• 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.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• 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.

• Stiffness matrix

If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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