- 9.5.1: is a triangular display of the binomial coefficients.
- 9.5.2: a b = n 0 b = and . an1 a b =
- 9.5.3: True or False anjb = j!1n - j2! n!
- 9.5.4: The can be used to expand expressions like 12x + 326 .
- 9.5.5: a 5 3 b
- 9.5.6: a 7 3 b
- 9.5.7: a 7 5 b
- 9.5.8: a 9 7 b
- 9.5.9: a 50 49 b
- 9.5.10: a 100 98 b
- 9.5.11: a 1000 1000 b
- 9.5.12: a 1000 0 b
- 9.5.13: a 55 23 b
- 9.5.14: a 60 20b
- 9.5.15: a 47 25 b
- 9.5.16: a 37 19 b
- 9.5.17: 1x + 125
- 9.5.18: 1x - 125
- 9.5.19: 1x - 226
- 9.5.20: 1x + 325
- 9.5.21: 13x + 124
- 9.5.22: 12x + 325
- 9.5.23: 1x2 + y2 2 5
- 9.5.24: 1x2 - y2 2 6
- 9.5.25: A 1x + 22B 6
- 9.5.26: A 1x - 23B 4
- 9.5.27: 1ax + by25
- 9.5.28: 1ax - by24
- 9.5.29: The coefficient of in the expansion of 1x + 3210
- 9.5.30: The coefficient of in the expansion of 1x - 3210
- 9.5.31: The coefficient of in the expansion of 12x - 1212
- 9.5.32: The coefficient of in the expansion of 12x + 1212
- 9.5.33: The coefficient of in the expansion of 12x + 32
- 9.5.34: The coefficient of in the expansion of 12x - 329
- 9.5.35: The fifth term in the expansion of 1x + 327
- 9.5.36: The third term in the expansion of 1x - 327
- 9.5.37: The third term in the expansion of 13x - 229
- 9.5.38: The sixth term in the expansion of 13x + 228
- 9.5.39: The coefficient of in the expansion of ax2 + 1 x b 12
- 9.5.40: The coefficient of in the expansion of x - 1 x2 9
- 9.5.41: The coefficient of in the expansion of ax - 2 1x b 10
- 9.5.42: The coefficient of in the expansion of a 1x + 3 1x b 8
- 9.5.43: Use the Binomial Theorem to find the numerical value of correct to ...
- 9.5.44: Use the Binomial Theorem to find the numerical value of correct to ...
- 9.5.45: Show that and ann a b = 1.
- 9.5.46: Show that if n and j are integers with then anjb = a nn - jb Conclu...
- 9.5.47: f n is a positive integer, show that [Hint: 2 now use the Binomial ...
- 9.5.48: If n is a positive integer, show that an0b - an1b + an2b - + 1-12n ...
- 9.5.49: a 5 0 b a 1 4 b 5 + a 5 1 b a 1 4 b 4 a 3 4 b + a 5 2 b a 1 4 b 3 a...
- 9.5.50: Stirlings Formula An approximation for n!, when n is large, is give...
Solutions for Chapter 9.5: The Binomial Theorem
Full solutions for College Algebra | 9th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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).
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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 •
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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!
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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