 43.1: Megan said that if a . 1 and g(x) 5 f(x), then the graph of f(x) is...
 43.2: Kyle said that if r is directly proportional to s, then there is so...
 43.3: In 36, each set represents a function. a. What is the domain of eac...
 43.4: In 36, each set represents a function. a. What is the domain of eac...
 43.5: In 36, each set represents a function. a. What is the domain of eac...
 43.6: In 36, each set represents a function. a. What is the domain of eac...
 43.7: In 712, determine if each graph represents a onetoone function
 43.8: In 712, determine if each graph represents a onetoone function
 43.9: In 712, determine if each graph represents a onetoone function
 43.10: In 712, determine if each graph represents a onetoone function
 43.11: In 712, determine if each graph represents a onetoone function
 43.12: In 712, determine if each graph represents a onetoone function
 43.13: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.14: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.15: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.16: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.17: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.18: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.19: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.20: In 1320: a. Graph each function. b. Is the function a direct variat...
 43.21: Is every linear function a direct variation?
 43.22: Is the direct variation of two variables always a linear function?
 43.23: In 2328, write an equation of the direct variation described. The c...
 43.24: In 2328, write an equation of the direct variation described. The d...
 43.25: In 2328, write an equation of the direct variation described. The l...
 43.26: In 2328, write an equation of the direct variation described. The w...
 43.27: In 2328, write an equation of the direct variation described. Water...
 43.28: In 2328, write an equation of the direct variation described. At 9:...
Solutions for Chapter 43: Linear Functions And Direct Variation
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 43: Linear Functions And Direct Variation
Get Full SolutionsSince 28 problems in chapter 43: Linear Functions And Direct Variation have been answered, more than 29184 students have viewed full stepbystep solutions from this chapter. Chapter 43: Linear Functions And Direct Variation includes 28 full stepbystep solutions. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. This expansive textbook survival guide covers the following chapters and their solutions.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

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

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.