- 4.1: The slope of the line between (2, 10) and (x2, 4) is 3. Find the va...
- 4.2: Give the slope and the y-intercept for each equation. a. y = 4 3x b...
- 4.3: Line a and line b are shown on the graph at right. Name the slope a...
- 4.4: Write each equation in the form requested. Check your answers by gr...
- 4.5: Consider the point-slope equation y = 3.5 + 2(x + 4.5). a. Name the...
- 4.6: Show all steps for a symbolic solution to each problem. a. 4 + 2.8x...
- 4.7: APPLICATION Suppose Karl bought a used car for $12,600. Each year i...
- 4.8: Recall the data about heating a pot of water from the investigation...
- 4.9: APPLICATION The table gives the winning heights for the Olympic wom...
- 4.10: APPLICATION This table shows the federal minimum hourly wage for 19...
- 4.11: Explain how to find the equation of a line when you know a. The slo...
Solutions for Chapter 4: Fitting a Line to Data
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Solvable system Ax = b.
The right side b is in the column space of A.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.