 2.3.1: To set up a model linear equation to fit realworld applications, w...
 2.3.2: Use your own words to describe this equation where n is a number: 5...
 2.3.3: If the total amount of money you had to invest was $2,000 and you d...
 2.3.4: If a man sawed a 10ft board into two sections and one section was ...
 2.3.5: If Bill was traveling v mi/h, how would you represent Daemons speed...
 2.3.6: For the following exercises, use the information to find a linear a...
 2.3.7: For the following exercises, use the information to find a linear a...
 2.3.8: For the following exercises, use the information to find a linear a...
 2.3.9: For the following exercises, use this scenario: Two different telep...
 2.3.10: For the following exercises, use this scenario: Two different telep...
 2.3.11: For the following exercises, use this scenario: Two different telep...
 2.3.12: For the following exercises, use this scenario: Two different telep...
 2.3.13: For the following exercises, use this scenario: A wireless carrier ...
 2.3.14: For the following exercises, use this scenario: A wireless carrier ...
 2.3.15: For the following exercises, use this scenario: A wireless carrier ...
 2.3.16: For the following exercises, use this scenario: A wireless carrier ...
 2.3.17: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.18: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.19: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.20: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.21: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.22: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.23: For exercises 17 and 18, use this scenario: A retired woman has $50...
 2.3.24: For the following exercises, use this scenario: A truck rental agen...
 2.3.25: For the following exercises, use this scenario: A truck rental agen...
 2.3.26: For the following exercises, use this scenario: A truck rental agen...
 2.3.27: For the following exercises, use this scenario: A truck rental agen...
 2.3.28: For the following exercises, find the slope of the lines that pass ...
 2.3.29: For the following exercises, find the slope of the lines that pass ...
 2.3.30: For the following exercises, find the slope of the lines that pass ...
 2.3.31: For the following exercises, find the slope of the lines that pass ...
 2.3.32: For the following exercises, solve for the given variable in the fo...
 2.3.33: For the following exercises, solve for the given variable in the fo...
 2.3.34: For the following exercises, solve for the given variable in the fo...
 2.3.35: For the following exercises, solve for the given variable in the fo...
 2.3.36: For the following exercises, solve for the given variable in the fo...
 2.3.37: For the following exercises, solve for the given variable in the fo...
 2.3.38: The area of a trapezoid is given by A = _ 1 2 h(b1 + b2 ). Use the ...
 2.3.39: Solve for h: A = _ 1 2 h(b1 + b2 )
 2.3.40: Use the formula from the previous question to find the height of a ...
 2.3.41: Find the dimensions of an American football field. The length is 20...
 2.3.42: Distance equals rate times time, d = rt. Find the distance Tom trav...
 2.3.43: Using the formula in the previous exercise, find the distance that ...
 2.3.44: What is the total distance that two people travel in 3 h if one of ...
 2.3.45: If the area model for a triangle is A = _ 1 2 bh, find the area of ...
 2.3.46: Solve for h: A = _ 1 2 bh
 2.3.47: Use the formula from the previous question to find the height to th...
 2.3.48: The volume formula for a cylinder is V = r2 h. Using the symbol in ...
 2.3.49: Solve for h: V = r2 h
 2.3.50: Use the formula from the previous question to find the height of a ...
 2.3.51: Solve for r: V = r2 h
 2.3.52: Use the formula from the previous question to find the radius of a ...
 2.3.53: The formula for the circumference of a circle is C = 2r. Find the c...
 2.3.54: Solve the formula from the previous question for . Notice why is so...
Solutions for Chapter 2.3: MODELS AND APPLICATIONS
Full solutions for College Algebra  1st Edition
ISBN: 9781938168383
Solutions for Chapter 2.3: MODELS AND APPLICATIONS
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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!

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.