- 2.1: Which statement below is true? A _3 5 < 0.60 < 6% B _3 5 = 0.06 = 6...
- 2.2: If 5 -1 = _1 5 and 5 -2 = _1 25 , what is the value of 5 -4 ? A _1 ...
- 2.3: Which symbol will make the number sentence true when placed in the ...
- 2.4: Kuri has to multiply -_3 7 by _4 5 to finish a math problem on her ...
- 2.5: What is the value of y in the equation -7y = 49? A y = -7 B y = -6 ...
- 2.6: Which expression could be used to find how many times -_1 6 goes in...
- 2.7: Which number below is not a rational number? A 0.0056 B 0. 6 C _2 3...
- 2.8: Robert correctly solves the problem -_9 12 _2 3 . What is his answe...
- 2.9: What is the value of x in the equation -x + 6 = 18? A x = -13 B x =...
- 2.10: What is the solution to 2 3 3 2 ? A 5 5 B 6 6 C 36 D 72
- 2.11: Ella and her friend Sonia hiked two different trails in Paris Mount...
- 2.12: Dimitri is building a bookshelf. On the instructions, he learns tha...
Solutions for Chapter 2: Math Connects: Concepts, Skills, and Problem Solving Course 3 0th Edition
Full solutions for Math Connects: Concepts, Skills, and Problem Solving Course 3 | 0th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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!
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
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].
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.
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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 •
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).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.