 5.1.1: Let A D 2 3 4 7 and B D 3 4 5 1 : Find (a) 2A C 3B; (b) 3A 2B; (c) ...
 5.1.2: Verify that (a) A.BC/ D .AB/C and that (b) A.BCC/ D AB C AC, where ...
 5.1.3: Find AB and BA given A D 2 0 1 3 4 5 and B D 2 4 1 3 7 0 3 2 3 5 :
 5.1.4: . Let A and B be the matrices given in and let x D 2t et and y D 2 ...
 5.1.5: . Let A and B be the matrices given in and let x D 2t et and y D 2 ...
 5.1.6: Let A1 D 2 1 3 2 ; A2 D 1 3 1 2 ; B D 2 4 1 2 : (a) Show that A1B D...
 5.1.7: Compute the determinants of the matrices A and B in 6. Are your res...
 5.1.8: Suppose that A and B are the matrices of 5. Verify that det.AB/ D d...
 5.1.9: A.t / D " t 2t 1 t3 1 t # and B.t / D 1 t 1 C t 3t2 4t3
 5.1.10: A.t / D 2 4 et t t2 t 02 8t 1 t3 3 5 and B.t / D 2 4 3 2et 3t 3 5
 5.1.11: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.12: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.13: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.14: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.15: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.16: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.17: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.18: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.19: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.20: In 11 through 20, write the given system in the form x0 D P.t /x C ...
 5.1.21: In 21 through 30, first verify that the given vectors are solutions...
 5.1.22: In 21 through 30, first verify that the given vectors are solutions...
 5.1.23: In 21 through 30, first verify that the given vectors are solutions...
 5.1.24: In 21 through 30, first verify that the given vectors are solutions...
 5.1.25: In 21 through 30, first verify that the given vectors are solutions...
 5.1.26: In 21 through 30, first verify that the given vectors are solutions...
 5.1.27: In 21 through 30, first verify that the given vectors are solutions...
 5.1.28: In 21 through 30, first verify that the given vectors are solutions...
 5.1.29: In 21 through 30, first verify that the given vectors are solutions...
 5.1.30: In 21 through 30, first verify that the given vectors are solutions...
 5.1.31: In 31 through 40, find a particular solution of the indicated linea...
 5.1.32: In 31 through 40, find a particular solution of the indicated linea...
 5.1.33: In 31 through 40, find a particular solution of the indicated linea...
 5.1.34: In 31 through 40, find a particular solution of the indicated linea...
 5.1.35: In 31 through 40, find a particular solution of the indicated linea...
 5.1.36: In 31 through 40, find a particular solution of the indicated linea...
 5.1.37: In 31 through 40, find a particular solution of the indicated linea...
 5.1.38: In 31 through 40, find a particular solution of the indicated linea...
 5.1.39: In 31 through 40, find a particular solution of the indicated linea...
 5.1.40: In 31 through 40, find a particular solution of the indicated linea...
 5.1.41: (a) Show that the vector functions x1.t / D t t2 and x2 D t2 t3 are...
 5.1.42: Suppose that one of the vector functions x1.t / D x11.t / x21.t / a...
 5.1.43: Suppose that the vectors x1.t / and x2.t / of are solutions of the ...
 5.1.44: Generalize 42 and 43 to prove Theorem 2 for n an arbitrary positive...
 5.1.45: Let x1.t /, x2.t /, :::; xn.t / be vector functions whose ith compo...
Solutions for Chapter 5.1: Matrices and Linear Systems
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 5.1: Matrices and Linear Systems
Get Full SolutionsChapter 5.1: Matrices and Linear Systems includes 45 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 45 problems in chapter 5.1: Matrices and Linear Systems have been answered, more than 15264 students have viewed full stepbystep solutions from this chapter. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

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!).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

lAII = 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 combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Outer product uv T
= column times row = rank one matrix.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose 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).

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.