 6.4.51: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.52: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.53: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.54: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.55: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.56: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.57: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.58: In Exercises 5158, write the partial fraction decomposition ofthe r...
 6.4.59: In Exercises 59 and 60, (a) writethe partial fraction decomposition...
 6.4.60: In Exercises 59 and 60, (a) writethe partial fraction decomposition...
 6.4.61: ENVIRONMENT The predicted cost (in thousandsof dollars) for a compa...
 6.4.62: THERMODYNAMICS The magnitude of the rangeof exhaust temperatures (i...
 6.4.63: TRUE OR FALSE? In Exercises 6365, determine whetherthe statement is...
 6.4.64: TRUE OR FALSE? In Exercises 6365, determine whetherthe statement is...
 6.4.65: TRUE OR FALSE? In Exercises 6365, determine whetherthe statement is...
 6.4.66: CAPSTONE Explain the similarities and differencesin finding the par...
 6.4.67: In Exercises 6770, write the partial fraction decompositionof the r...
 6.4.68: In Exercises 6770, write the partial fraction decompositionof the r...
 6.4.69: In Exercises 6770, write the partial fraction decompositionof the r...
 6.4.70: In Exercises 6770, write the partial fraction decompositionof the r...
Solutions for Chapter 6.4: Partial Fractions
Full solutions for College Algebra  8th Edition
ISBN: 9781439048696
Solutions for Chapter 6.4: Partial Fractions
Get Full SolutionsCollege Algebra was written by and is associated to the ISBN: 9781439048696. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: College Algebra , edition: 8. Chapter 6.4: Partial Fractions includes 20 full stepbystep solutions. Since 20 problems in chapter 6.4: Partial Fractions have been answered, more than 33420 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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).

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

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.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.