 8.8.1.142: The vector is a solution of for k __________.
 8.8.1.143: . The vector is solution of the initialvalue problem for c1 ______...
 8.8.1.144: Consider the linear system . Without attempting to solve the system...
 8.8.1.145: Consider the linear system X AX of two differential equations, wher...
 8.8.1.146: In 514 solve the given linear system.dx dt 2x y
 8.8.1.147: In 514 solve the given linear system.dx dt 4x 2y
 8.8.1.148: In 514 solve the given linear system.X X 1 2 2 1X dy
 8.8.1.149: In 514 solve the given linear system.X 2254X X
 8.8.1.150: In 514 solve the given linear system.X X104113131X
 8.8.1.151: In 514 solve the given linear system.X 012212121 X X
 8.8.1.152: In 514 solve the given linear system.X 2 0 8 4X 2 16t
 8.8.1.153: In 514 solve the given linear system.X 11221X 0ettan t
 8.8.1.154: In 514 solve the given linear system.X 1211X 1cot t
 8.8.1.155: In 514 solve the given linear system.X 3111X 21e2t
 8.8.1.156: (a) Consider the linear system X AX of three first order differenti...
 8.8.1.157: Verify that is a solution of the linear system for arbitrary consta...
Solutions for Chapter 8: Systems of Linear FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 8: Systems of Linear FirstOrder Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Since 16 problems in chapter 8: Systems of Linear FirstOrder Differential Equations have been answered, more than 21385 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8: Systems of Linear FirstOrder Differential Equations includes 16 full stepbystep solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination.
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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to 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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.