- 2.5.1: Rewrite each equation in Y = form. a. xy = 15 b. xy = 35 c. xy = 3
- 2.5.2: Two quantities, x and y, are inversely proportional. When x = 3, y ...
- 2.5.3: Find five points that satisfy the inverse variation equation Graph ...
- 2.5.4: Henry noticed that the more television he atched, the less time he ...
- 2.5.5: APPLICATION The amount of time it takes to travel a given distance ...
- 2.5.6: For each table of x- and y-values below, decide if the values show ...
- 2.5.7: APPLICATION In Example A, you learned that the force in newtons nee...
- 2.5.8: APPLICATION Emily and her little brother Sid are playing on a seesa...
- 2.5.9: To use a double-pan balance, you put the object to be weighed on on...
- 2.5.10: APPLICATION The student council wants to raise $10,000 to purchase ...
- 2.5.11: A tuning fork vibrates at a particular frequency to make the sound ...
- 2.5.12: APPLICATION To squeeze a given amount of air into a smaller and sma...
- 2.5.13: APPLICATION A CD is on sale for 15% off its normal price of $13.95....
- 2.5.14: Calcium and phosphorus play important roles in building human bones...
- 2.5.15: APPLICATION Two dozen units in an apartment complex need to be pain...
- 2.5.16: Sulfuric acid, a highly corrosive substance, is used in the manufac...
Solutions for Chapter 2.5: Inverse Variation
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
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.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
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).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).