 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
ISBN: 9780078740503
Solutions for Chapter 2
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Back substitution.
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!

Cyclic shift
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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Graph G.
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.

Kirchhoff's Laws.
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.

Lucas numbers
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 .

Multiplication Ax
= 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.

Normal matrix.
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+ (MoorePenrose 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.

Schwarz inequality
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.