- 10.19.1: Find the trigonometric polynomial of order 3 that is the least squa...
- 10.19.2: Find the trigonometric polynomial of order 4 that is the least squa...
- 10.19.3: Find the trigonometric polynomial of order 4 that is the least squa...
- 10.19.4: Find the trigonometric polynomial of arbitrary order n that is the ...
- 10.19.5: Find the trigonometric polynomial of arbitrary order n that is the ...
- 10.19.6: For the inner product show that (a) (b) (c)
- 10.19.7: Show that the functions are orthogonal over the interval relative t...
- 10.19.8: If is defined and continuous on the interval , show that is defined...
- 10.19.T1: Let g be the function for . Use a computer to determine the Fourier...
- 10.19.T2: Let g be the function for . Use a computer to determine the Fourier...
Solutions for Chapter 10.19: A Least Squares Model for Human Hearing
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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.
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.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.