- 10.12.1: The self-similar set in Figure Ex-1 has the sizes indicated. Given ...
- 10.12.2: Find the Hausdorff dimension of the self-similar set shown in Figur...
- 10.12.3: Each of the 12 self-similar sets in Figure Ex-3 results from three ...
- 10.12.4: For each of the self-similar sets in Figure Ex-4, find: (i) the sca...
- 10.12.5: Show that of the four affine transformations shown in Figure 10.12....
- 10.12.6: Find the coordinates of the tip of the fern in Figure 10.12.22. [Hi...
- 10.12.7: The square in Figure 10.12.7a was expressed as the union of 4 nonov...
- 10.12.8: Show that the four similitudes T1 x y = 3 4 1 0 0 1 x y T2 x y = 3 ...
- 10.12.9: All of the results in this section can be extended to Rn. Compute t...
- 10.12.10: The set in R3 in Figure Ex-10 is called the Menger sponge. It is a ...
- 10.12.11: The two similitudes T1 x y = 1 3 1 0 0 1 x y and T2 x y = 1 3 1 0 0...
- 10.12.12: Compute the areas of the sets S0, S1, S2, S3, and S4 in Figure 10.1...
- 10.12.T1: Use similitudes of the form Ti x y z = 1 3 100 010 001 x y z + ai b...
- 10.12.T2: Generalize the ideas involved in the Cantor set (in R1), the Sierpi...
Solutions for Chapter 10.12: Fractals
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version | 11th 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!
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
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].
Invert A by row operations on [A I] to reach [I A-I].
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.
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.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(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!
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.