 1.7.1E: In Exercises 1–4, determine if the vectors are linearly independent...
 1.7.2E: Determine if the vectors are linearly independent. Justify each ans...
 1.7.3E: Determine if the vectors are linearly independent. Justify each ans...
 1.7.4E: Determine if the vectors are linearly independent. Justify each ans...
 1.7.5E: Determine if the columns of the matrix form a linearly independent ...
 1.7.6E: Determine if the columns of the matrix form a linearly independent ...
 1.7.7E: In Exercises 5–8, determine if the columns of the matrix form a lin...
 1.7.8E: Determine if the columns of the matrix form a linearly independent ...
 1.7.9E: In Exercises 9 and 10, (a) for what values of h is v3 in Span {v1, ...
 1.7.10E: (a) for what values of h is v3 in Span {v1, v2}, and (b) for what v...
 1.7.11E: Find the value(s) of h for which the vectors are linearly dependent...
 1.7.12E: Find the value(s) of h for which the vectors are linearly dependent...
 1.7.13E: In Exercises 11–14, find the value(s) of h for which the vectors ar...
 1.7.14E: Find the value(s) of h for which the vectors are linearly dependent...
 1.7.15E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.16E: Determine by inspection whether the vectors in Exercises are linear...
 1.7.17E: Determine by inspection whether the vectors in Exercises are linear...
 1.7.18E: Determine by inspection whether the vectors in Exercises are linear...
 1.7.19E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.20E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 1.7.22E: a. Two vectors are linearly dependent if and only if they lie on a ...
 1.7.23E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.24E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.25E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.26E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.27E: How many pivot columns must a 7 × 5 matrix have if its columns are ...
 1.7.28E: How many pivot columns must a 5 × 7 matrix have if its columns span...
 1.7.29E: Construct 3 × 2 matrices A and B such that Ax = 0 has a nontrivial ...
 1.7.30E: a. Fill in the blank in the following statement: “If A is an m × n ...
 1.7.31E: Exercises 31 and 32 should be solved without performing row operati...
 1.7.32E: Given A = observe that the first column plus twice the second colum...
 1.7.33E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.34E: If v1,...,v4 are in ?4 and v3 =0, then {v1, v2, v3, v4} is linearly...
 1.7.35E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.36E: If v1,..., v4 are in ?4 and v3 is not a linear combination of v1, v...
 1.7.37E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.38E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.39E: Suppose A is an m × n matrix with the property that for all b in ?m...
 1.7.40E: Suppose an m × n matrix A has n pivot columns. Explain why for each...
 1.7.41E: Use as many columns of A as possible to construct a matrix B with t...
 1.7.42E: Use as many columns of A as possible to construct a matrix B with t...
 1.7.43E: [M] With A and B as in Exercise 41, select a column v of A that was...
 1.7.44E: [M] Repeat Exercise 43 with the matrices A and B from Exercise 42. ...
Solutions for Chapter 1.7: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 1.7
Get Full SolutionsChapter 1.7 includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5th. Since 44 problems in chapter 1.7 have been answered, more than 22957 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. 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).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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).

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.