- 1.4.1E: (a) set up the system of equations that describes traffic flow; (b)...
- 1.4.2E: (a) set up the system of equations that describes traffic flow; (b)...
- 1.4.3E: find the flow of traffic in the rotary if x1= 600.
- 1.4.4E: find the flow of traffic in the rotary if x1= 600.
- 1.4.5E: determine the currents in the various branches.
- 1.4.6E: determine the currents in the various branches.
- 1.4.7E: determine the currents in the various branches.
- 1.4.8E: determine the currents in the various branches.
- 1.4.9E: a) Set up (he system of equations that describes the traffic Mow in...
- 1.4.10E: The electrical network shown in the accompanying figure is called a...
Solutions for Chapter 1.4: Introduction to Linear Algebra 5th Edition
Full solutions for Introduction to Linear Algebra | 5th Edition
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!
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
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.
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 ().
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.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.