 3.SE.1E: Mark each statement True or False. Justify each answer. Assume that...
 3.SE.2E: Use row operations to show that the determinants in Exercises 2–4 a...
 3.SE.3E: Use row operations to show that the determinants in Exercises 2–4 a...
 3.SE.4E: Use row operations to show that the determinants in Exercises 2–4 a...
 3.SE.5E: Compute the determinants in Exercises 5 and 6.919999099240050903906...
 3.SE.6E: Compute the determinants in Exercises 5 and 6.488850100068887088300...
 3.SE.7E: Show that the equation of the line in can be written as
 3.SE.8E: Find a 3 × 3 determinant equation similar to that in Exercise 7 tha...
 3.SE.9E: Exercises 9 and 10 concern determinants of the following Vandermond...
 3.SE.10E: Exercises 9 and 10 concern determinants of the following Vandermond...
 3.SE.11E: Determine the area of the parallelogram determined by the points Ho...
 3.SE.12E: Use the concept of area of a parallelogram to write a statement abo...
 3.SE.13E: Show that if A is invertible, then adj A is invertible, and [Hint: ...
 3.SE.14E: Let A, B, C, D, and I be n × n matrices. Use the definition or prop...
 3.SE.15E: Let A, B, C, and D be n × n matrices with A invertible.a. Find matr...
 3.SE.16E: Let J be the n × n matrix of all 1’s, and consider a. Subtract row ...
 3.SE.17E: Let A be the original matrix given in Exercise 16, and let Notice t...
 3.SE.18E: [M] Apply the result of Exercise 16 to find the determinants of the...
 3.SE.19E: [M] Use a matrix program to compute the determinants of the followi...
 3.SE.20E: [M] Use the method of Exercise 19 to guess the determinant of Justi...
Solutions for Chapter 3.SE: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 3.SE
Get Full SolutionsSince 20 problems in chapter 3.SE have been answered, more than 30226 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Chapter 3.SE includes 20 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

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

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

Outer product uv T
= column times row = rank one matrix.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Solvable system Ax = b.
The right side b is in the column space of A.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.