 25.1: If , then is ? Justify your answer.
 25.2: If , then is ? Justify your answer.
 25.3: In 310, write each ratio in simplest form. 12 : 8
 25.4: In 310, write each ratio in simplest form. 21 : 14
 25.5: In 310, write each ratio in simplest form. 3 : 18
 25.6: In 310, write each ratio in simplest form. 15 : 75
 25.7: In 310, write each ratio in simplest form. 6a : 9a, a 0
 25.8: In 310, write each ratio in simplest form. 18 27
 25.9: In 310, write each ratio in simplest form. 24 72
 25.10: In 310, write each ratio in simplest form. 10x 35x x 0
 25.11: In 1119, solve each proportion for the variable.
 25.12: In 1119, solve each proportion for the variable.
 25.13: In 1119, solve each proportion for the variable.
 25.14: In 1119, solve each proportion for the variable.
 25.15: In 1119, solve each proportion for the variable.
 25.16: In 1119, solve each proportion for the variable.
 25.17: In 1119, solve each proportion for the variable.
 25.18: In 1119, solve each proportion for the variable.
 25.19: In 1119, solve each proportion for the variable.
 25.20: The ratio of the length to the width of a rectangle is 5 : 4. The p...
 25.21: The ratio of the length to the width of a rectangle is 7 : 3. The a...
 25.22: The basketball team has played 21 games. The ratio of wins to losse...
 25.23: In the chess club, the ratio of boys to girls is 6 : 5. There are 3...
 25.24: Every year, Javier makes a total contribution of $125 to two local ...
 25.25: A cookie recipe uses flour and sugar in the ratio of 9 : 4. If Nich...
 25.26: The directions on a bottle of cleaning solution suggest that the so...
Solutions for Chapter 25: Ratio And Proportion
Full solutions for Amsco's Algebra 2 and Trigonometry  1st Edition
ISBN: 9781567657029
Solutions for Chapter 25: Ratio And Proportion
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 26 problems in chapter 25: Ratio And Proportion have been answered, more than 30871 students have viewed full stepbystep solutions from this chapter. Chapter 25: Ratio And Proportion includes 26 full stepbystep solutions. Amsco's Algebra 2 and Trigonometry was written by and is associated to the ISBN: 9781567657029. This textbook survival guide was created for the textbook: Amsco's Algebra 2 and Trigonometry, edition: 1.

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.)

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Pascal matrix
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).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).