- 10.3.1: APPLICATION Suppose there are 180 twelfth graders in your school, a...
- 10.3.2: APPLICATION Last month it was estimated that a lake contained 3500 ...
- 10.3.3: Suppose 250 people have applied for 15 job openings at a chain of r...
- 10.3.4: If 25 randomly plotted points landed in the shaded region shown in ...
- 10.3.5: In a random walk, you move according to rules with each move being ...
- 10.3.6: A thumbtack can land point up or point down. a. When you drop a thu...
- 10.3.7: APPLICATION A teacher would like to use her calculator to randomly ...
- 10.3.8: Use algebraic techniques to solve 8a and b. Then use calculator gra...
- 10.3.9: APPLICATION Zoe is an intern at Yellowstone National Park. One of h...
- 10.3.10: For 10af, if the number has an exponent, write it in standard form....
- 10.3.11: Explain how to use probability to find the area of the irregular sh...
Solutions for Chapter 10.3: Random Outcomes
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column space C (A) =
space of all combinations of the columns of A.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Invert A by row operations on [A I] to reach [I A-I].
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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).
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.