 7.5.1: Exer. 18: Express as a sum or difference
 7.5.2: Exer. 18: Express as a sum or difference
 7.5.3: Exer. 18: Express as a sum or difference
 7.5.4: Exer. 18: Express as a sum or difference
 7.5.5: Exer. 18: Express as a sum or difference
 7.5.6: Exer. 18: Express as a sum or difference
 7.5.7: Exer. 18: Express as a sum or difference
 7.5.8: Exer. 18: Express as a sum or difference
 7.5.9: Exer. 916: Express as a product.
 7.5.10: Exer. 916: Express as a product.
 7.5.11: Exer. 916: Express as a product.
 7.5.12: Exer. 916: Express as a product.
 7.5.13: Exer. 916: Express as a product.
 7.5.14: Exer. 916: Express as a product.
 7.5.15: Exer. 916: Express as a product.
 7.5.16: Exer. 916: Express as a product.
 7.5.17: Exer. 1724: Verify the identity
 7.5.18: Exer. 1724: Verify the identity
 7.5.19: Exer. 1724: Verify the identity
 7.5.20: Exer. 1724: Verify the identity
 7.5.21: Exer. 1724: Verify the identity
 7.5.22: Exer. 1724: Verify the identity
 7.5.23: Exer. 1724: Verify the identity
 7.5.24: Exer. 1724: Verify the identity
 7.5.25: Exer. 2526: Express as a sum.
 7.5.26: Exer. 2526: Express as a sum.
 7.5.27: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.28: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.29: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.30: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.31: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.32: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.33: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.34: Exer. 2734: Use sumtoproduct formulas to find the solutions of th...
 7.5.35: Exer. 3536: Shown in the figure is a graph of the function f for 0 ...
 7.5.36: Exer. 3536: Shown in the figure is a graph of the function f for 0 ...
 7.5.37: Refer to Exercise 47 of Section 7.4. The graph of the equation has ...
 7.5.38: Refer to Exercise 48 of Section 7.4. The xcoordinates of the turni...
 7.5.39: Mathematical analysis of a vibrating violin string of length l invo...
 7.5.40: If two tuning forks are struck simultaneously with the same force a...
Solutions for Chapter 7.5: Productt oSum and Sumt oProduct For mulas
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 7.5: Productt oSum and Sumt oProduct For mulas
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 40 problems in chapter 7.5: Productt oSum and Sumt oProduct For mulas have been answered, more than 37531 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Chapter 7.5: Productt oSum and Sumt oProduct For mulas includes 40 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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 nl  An).

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

GramSchmidt 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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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·