- 103.1: When the sum of 12 and 13 is divided by the product of 12 and 13 wh...
- 103.2: The average age of three men is 24 years. a. What is the sum of the...
- 103.3: A string one yard long is formed into the shape of a square. a. How...
- 103.4: Complete this proportion: Five is to three as thirty is to whatnumber?
- 103.5: Mr. Cho has 30 books. Fourteen of the books are mysteries. What is ...
- 103.6: In another class of 33 students, the ratio of boys to girls is4 to ...
- 103.7: 100 102 3 (23 216)
- 103.8: Robert complained that he had a ton of homework. a. How many pounds...
- 103.9: Complete the table to answer problems 911.1100 a. b.
- 103.10: Complete the table to answer problems 911.a. 0.4 b.
- 103.11: Complete the table to answer problems 911.a. b. 8%
- 103.12: 1012 312
- 103.13: (6 + 2.4) 0.04
- 103.14: Find each unknown number:712 634 n 1538
- 103.15: Find each unknown number:x 134 712
- 103.16: Instead of dividing 1012 by 312, Guadalupe doubled both numbersbefo...
- 103.17: Mariabella used a tape measure to find the circumferenceand the dia...
- 103.18: Write twenty million, five hundred thousand in expanded notation us...
- 103.19: Name the prime numbers between 40 and 50.
- 103.20: Calculate mentally: a. 3 + 8 b. 3 8 c. 8 + +3 d. 8 +3
- 103.21: In ABC the measure of A is40. Angles B and C are congruent. What is...
- 103.22: a. What is the perimeter of this triangle? b. What is the area of t...
- 103.23: The Simpsons rented a trailer that was 8 feet long and 5 feetwide. ...
- 103.24: What is the probability of drawing the queen of spades from a norma...
- 103.25: a. What temperature is shown onthis thermometer? b. If the temperat...
- 103.26: Find the perimeter of the figure below.
- 103.27: a. What is the area of the shadedrectangle? b. What is the area of ...
- 103.28: What are the coordinates of the point halfway between(3, 2) and (5,...
- 103.29: A pint of milk weighs about 16 ounces. About how manypounds does a ...
- 103.30: Ruben walked around a building whose perimeter was shapedlike a reg...
Solutions for Chapter 103: Perimeter of Complex Shapes
Full solutions for Saxon Math, Course 1 | 1st Edition
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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].
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.
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.
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 .
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Every v in V is orthogonal to every w in W.
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 •
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.