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# Solutions for Chapter 1.3: Solving Equations

## Full solutions for California Algebra 2: Concepts, Skills, and Problem Solving | 1st Edition

ISBN: 9780078778568

Solutions for Chapter 1.3: Solving Equations

Solutions for Chapter 1.3
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##### ISBN: 9780078778568

Since 83 problems in chapter 1.3: Solving Equations have been answered, more than 42178 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.3: Solving Equations includes 83 full step-by-step solutions. This textbook survival guide was created for the textbook: California Algebra 2: Concepts, Skills, and Problem Solving, edition: 1. California Algebra 2: Concepts, Skills, and Problem Solving was written by and is associated to the ISBN: 9780078778568.

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

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Cholesky factorization

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

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Cramer's Rule for Ax = b.

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

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Eigenvalue A and eigenvector x.

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

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

Use AT for complex A.

• Free columns of A.

Columns without pivots; these are combinations of earlier columns.

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• Fundamental Theorem.

The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

• Hermitian matrix A H = AT = A.

Complex analog a j i = aU of a symmetric matrix.

• Length II x II.

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

• 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.

• 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).

• Polar decomposition A = Q H.

Orthogonal Q times positive (semi)definite H.

• 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.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

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