 1.1.1E: Solve each system in Exercises 1–4 by using elementary row operatio...
 1.1.2E: Solve each system in Exercises 1–4 by using elementary row operatio...
 1.1.3E: Solve each system in Exercises 1–4 by using elementary row operatio...
 1.1.4E: Solve each system in Exercises 1–4 by using elementary row operatio...
 1.1.5E: Consider each matrix in Exercises 5 and 6 as the augmented matrix o...
 1.1.6E: Consider each matrix in Exercises 5 and 6 as the augmented matrix o...
 1.1.7E: In Exercises 7–10, the augmented matrix of a linear system has been...
 1.1.8E: In Exercises 7–10, the augmented matrix of a linear system has been...
 1.1.9E: In Exercises 7–10, the augmented matrix of a linear system has been...
 1.1.10E: In Exercises 7–10, the augmented matrix of a linear system has been...
 1.1.11E: Solve the systems in Exercises 11–14.x2 + 5x3 = –4x1 + 4x2 + 3x3 = ...
 1.1.12E: Solve the systems in Exercises 11–14.x1 – 5x2 + 4x3 = –32x1 – 7x2 +...
 1.1.13E: Solve the systems in Exercises 11–14.x1 –3x3 = 82x1 + 2x2 + 9x3 = 7...
 1.1.14E: Solve the systems in Exercises 11–14.2x1 –6x3 = –8 x2 + 2x3 = 33x1 ...
 1.1.15E: Determine if the systems in Exercises 15 – 16 are consistent. Do no...
 1.1.16E: Determine if the systems in Exercises 15 – 16 are consistent. Do no...
 1.1.17E: Do the three lines 2x1 + 3x2 = –1, 6x1 + 5x2 = 0 and 2x1 – 5x2 = 7 ...
 1.1.18E: Do the three planes 2x1 + 4x2 + 4x3 = 4, x2 – 2x3 = – 2 and 2x1 + 3...
 1.1.19E: In Exercises 19–22, determine the value(s) of h such that the matri...
 1.1.20E: In Exercises 19–22, determine the value(s) of h such that the matri...
 1.1.21E: In Exercises 19–22, determine the value(s) of h such that the matri...
 1.1.22E: In Exercises 19–22, determine the value(s) of h such that the matri...
 1.1.23E: In Exercises 23 and 24, key statements from this section are either...
 1.1.24E: In Exercises 23 and 24, key statements from this section are either...
 1.1.25E: Find an equation involving g, h, and k that makes this augmented ma...
 1.1.26E: Suppose the system below is consistent for all possible values of f...
 1.1.27E: Suppose a, b, c, and d are constants such that a is not zero and th...
 1.1.28E: Construct three different augmented matrices for linear systems who...
 1.1.29E: In Exercises 29–32, find the elementary row operation that transfor...
 1.1.30E: In Exercises 29–32, find the elementary row operation that transfor...
 1.1.31E: In Exercises 29–32, find the elementary row operation that transfor...
 1.1.32E: In Exercises 29–32, find the elementary row operation that transfor...
 1.1.33E: An important concern in the study of heat transfer is to determine ...
 1.1.34E: An important concern in the study of heat transfer is to determine ...
Solutions for Chapter 1.1: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 1.1
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Chapter 1.1 includes 34 full stepbystep solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Since 34 problems in chapter 1.1 have been answered, more than 32454 students have viewed full stepbystep solutions from this chapter.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.