- 1.10.1E: The container of a breakfast cereal usually lists the number of cal...
- 1.10.2E: In a certain region, about 6% of a city’s population moves to the s...
- 1.10.3E: [M] Budget® Rent A Car in Wichita, Kansas has a fleet of about 500 ...
- 1.10.4E: Clock’s moving hands At what rate is the angle between a clock’s mi...
- 1.10.5E: Oil spill An explosion at an oil rig located in gulf waters causes ...
- 1.10.6E: Find the first four terms of the Taylor series for the functions in...
- 1.10.7E: In Exercises 5–8, write a matrix equation that determines the loop ...
- 1.10.8E: In Exercise, use series to estimate the integrals’ values with an e...
- 1.10.9E: In Exercise, use series to estimate the integrals’ values with an e...
- 1.10.10E: In Exercise, use series to estimate the integrals’ values with an e...
- 1.10.11E: In 2012 the population of California was 38,041,430, and the popula...
- 1.10.12E: In Exercise, use series to estimate the integrals’ values with an e...
- 1.10.13E: [M] Let M and x0 be as in Example 3.a. Compute the population vecto...
- 1.10.14E: The complete elliptic integral of the first kind is the integral ,w...
Solutions for Chapter 1.10: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications | 5th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
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.
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.
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
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.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).