- 1.1.1: Use back substitution to solve each of the following systems of equ...
- 1.1.2: Write out the coefficient matrix for each of the systems in Exercis...
- 1.1.3: In each of the following systems, interpret each equation as a line...
- 1.1.4: Write an augmented matrix for each of the systems in Exercise 3.
- 1.1.5: Write out the system of equations that corresponds to each of the f...
- 1.1.6: Solve each of the following systems: (a) x1 2x2 = 5 3x1 + x2 = 1 (b...
- 1.1.7: The two systems 2x1 + x2 = 3 4x1 + 3x2 = 5 and 2x1 + x2 = 1 4x1 + 3...
- 1.1.8: Solve the two systems x1 + 2x2 2x3 = 1 2x1 + 5x2 + x3 = 9 x1 + 3x2 ...
- 1.1.9: Given a system of the form m1x1 + x2 = b1 m2x1 + x2 = b2 where m1, ...
- 1.1.10: Consider a system of the form a11x1 + a12x2 = 0 a21x1 + a22x2 = 0 w...
- 1.1.11: Give a geometrical interpretation of a linear equation in three unk...
Solutions for Chapter 1.1: Systems of Linear Equations
Full solutions for Linear Algebra with Applications | 8th Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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).
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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 ().
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.