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# 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

Solutions for Chapter 1.1
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##### ISBN: 9780321385178

This 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 step-by-step 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 step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• 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).

• lA-II = 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. A-I 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.

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