- 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 sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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).
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (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.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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