- 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
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!
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
Invert A by row operations on [A I] to reach [I A-I].
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 .
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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.
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 •
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).