 1.7.1E: In Exercises 1–4, determine if the vectors are linearly independent...
 1.7.2E: In Exercises 1–4, determine if the vectors are linearly independent...
 1.7.3E: In Exercises 1–4, determine if the vectors are linearly independent...
 1.7.4E: In Exercises 1–4, determine if the vectors are linearly independent...
 1.7.5E: In Exercises 5–8, determine if the columns of the matrix form a lin...
 1.7.6E: In Exercises 5–8, determine if the columns of the matrix form a lin...
 1.7.7E: In Exercises 5–8, determine if the columns of the matrix form a lin...
 1.7.8E: In Exercises 5–8, determine if the columns of the matrix form a lin...
 1.7.9E: In Exercises 9 and 10, (a) for what values of h is v3 in Span {v1, ...
 1.7.10E: In Exercises 9 and 10, (a) for what values of h is v3 in Span {v1, ...
 1.7.11E: In Exercises 11–14, find the value(s) of h for which the vectors ar...
 1.7.12E: In Exercises 11–14, find the value(s) of h for which the vectors ar...
 1.7.13E: In Exercises 11–14, find the value(s) of h for which the vectors ar...
 1.7.14E: In Exercises 11–14, find the value(s) of h for which the vectors ar...
 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 15–20 are ...
 1.7.17E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.18E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 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: In Exercises 21 and 22, mark each statement True or False. Justify ...
 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 6 × 4 matrix have if its columns are ...
 1.7.28E: How many pivot columns must a 4 × 6 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: Exercises 31 and 32 should be solved without performing row operati...
 1.7.33E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.34E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.35E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.36E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 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: [M] In Exercises 41 and 42, use as many columns of A as possible to...
 1.7.42E: [M] In Exercises 41 and 42, use as many columns of A as possible to...
 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 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 1.7
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Chapter 1.7 includes 44 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 44 problems in chapter 1.7 have been answered, more than 32527 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178.

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).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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!).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

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.