- 3.4.1: Match the recursive routine in the first column with the equation i...
- 3.4.2: You can use the equation d = 24 45t to model the distance from a de...
- 3.4.3: You can use the equation d = 4.7 + 2.8t to model a walk in which th...
- 3.4.4: Undo the order of operations to find the x-value in each equation. ...
- 3.4.5: The equation y = 35 + 0.8x gives the distance a sports car is from ...
- 3.4.6: APPLICATION Louis is beginning a new exercise workout. His trainer ...
- 3.4.7: Jo mows lawns after school. She finds that she can use the equation...
- 3.4.8: As part of a physics experiment, June threw an object off a cliff a...
- 3.4.9: APPLICATION Manny has a part-time job as a waiter. He makes $45 per...
- 3.4.10: APPLICATION Paula is cross-training for a triathlon in which she cy...
- 3.4.11: At a family picnic, your cousin tells you that he always has a hard...
- 3.4.12: APPLICATION Carl has been keeping a record of his gas purchases for...
- 3.4.13: Match each recursive routine to a graph below. Explain how you made...
- 3.4.14: Bjarne is training for a bicycle race by riding on a stationary bic...
- 3.4.15: Consider the expression . a. Find the value of the expression if y ...
Solutions for Chapter 3.4: Linear Equations and the Intercept Form
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } 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.
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.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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 .
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Every v in V is orthogonal to every w in W.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Solvable system Ax = b.
The right side b is in the column space of A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.