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# Solutions for Chapter 5.1: Linear Algebra and Its Applications 4th Edition

## Full solutions for Linear Algebra and Its Applications | 4th Edition

ISBN: 9780321385178

Solutions for Chapter 5.1

Solutions for Chapter 5.1
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##### ISBN: 9780321385178

This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Since 40 problems in chapter 5.1 have been answered, more than 80216 students have viewed full step-by-step solutions from this chapter. Chapter 5.1 includes 40 full step-by-step solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions.

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

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• 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.

• 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.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Outer product uv T

= column times row = rank one matrix.

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Rotation matrix

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 ().

• Sum V + W of subs paces.

Space of all (v in V) + (w in W). Direct sum: V n W = to}.

• Trace of A

= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

• Vandermonde matrix V.

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

• 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.

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.