- 6.2.1: Find the general solution of each of the following systems: (a) y_ ...
- 6.2.2: Solve each of the following initial value problems: (a) y_ 1 = y1 +...
- 6.2.3: Given Y = c1e1tx1 + c2e2tx2 + +cnen txn is the solution to the init...
- 6.2.4: Two tanks each contain 100 liters of a mixture. Initially, the mixt...
- 6.2.5: Find the general solution of each of the following systems: (a) y__...
- 6.2.6: Solve the initial value problem y__ 1 = 2y2 + y_ 1 + 2y_ 2 y__ 2 = ...
- 6.2.7: In Application 2, assume that the solutions are of the form x1 = a1...
- 6.2.8: Solve the the problem in Application 2, using the initial condition...
- 6.2.9: Two masses are connected by springs as shown in the accompanying di...
- 6.2.10: Three masses are connected by a series of springs between two fixed...
- 6.2.11: Transform the nth-order equation y(n) = a0 y + a1 y_ + +an1 y(n1) i...
Solutions for Chapter 6.2: Systems of Linear Differential Equations
Full solutions for Linear Algebra with Applications | 8th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Tv = Av + Vo = linear transformation plus shift.
peA) = det(A - AI) has peA) = zero matrix.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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).
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.
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.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.