 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...
 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  9th Edition
ISBN: 9780321962218
Solutions for Chapter 1.1: Systems of Linear Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. Chapter 1.1: Systems of Linear Equations includes 11 full stepbystep solutions. Since 11 problems in chapter 1.1: Systems of Linear Equations have been answered, more than 10891 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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).

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.