 3.3.1E: Use Cramer’s rule to compute the solutions of the systems in Exerci...
 3.3.2E: Use Cramer’s rule to compute the solutions of the systems in Exerci...
 3.3.3E: Use Cramer’s rule to compute the solutions of the systems in Exerci...
 3.3.4E: Use Cramer’s rule to compute the solutions of the systems in Exerci...
 3.3.5E: Use Cramer’s rule to compute the solutions of the systems in Exerci...
 3.3.6E: Use Cramer’s rule to compute the solutions of the systems in Exerci...
 3.3.7E: In Exercises 7–10, determine the values of the parameter s for whic...
 3.3.8E: In Exercises 7–10, determine the values of the parameter s for whic...
 3.3.9E: In Exercises 7–10, determine the values of the parameter s for whic...
 3.3.10E: In Exercises 7–10, determine the values of the parameter s for whic...
 3.3.11E: In Exercises 11–16, compute the adjugate of the given matrix, and t...
 3.3.12E: In Exercises 11–16, compute the adjugate of the given matrix, and t...
 3.3.13E: In Exercises 11–16, compute the adjugate of the given matrix, and t...
 3.3.14E: In Exercises 11–16, compute the adjugate of the given matrix, and t...
 3.3.15E: In Exercises 11–16, compute the adjugate of the given matrix, and t...
 3.3.16E: In Exercises 11–16, compute the adjugate of the given matrix, and t...
 3.3.17E: Show that if A is 2 × 2, then Theorem 8 gives the same formula for ...
 3.3.18E: Suppose that all the entries in A are integers and det A = 1. Expla...
 3.3.19E: In Exercises 19–22, find the area of the parallelogram whose vertic...
 3.3.20E: In Exercises 19–22, find the area of the parallelogram whose vertic...
 3.3.21E: In Exercises 19–22, find the area of the parallelogram whose vertic...
 3.3.22E: In Exercises 19–22, find the area of the parallelogram whose vertic...
 3.3.23E: Find the volume of the parallelepiped with one vertex at the origin...
 3.3.24E: Find the volume of the parallelepiped with one vertex at the origin...
 3.3.25E: Use the concept of volume to explain why the determinant of a 3 × 3...
 3.3.26E: be a linear transformation, and let p be a vector and S a set in Sh...
 3.3.27E: Let S be the parallelogram determined by the vectors .Compute the a...
 3.3.28E: Repeat Exercise 27 with Exercise 27:Let S be the parallelogram dete...
 3.3.29E: Find a formula for the area of the triangle whose vertices are
 3.3.30E: Let R be the triangle with vertices at Show that [Hint: Translate R...
 3.3.31E: Let be the linear transformation determined by the matrix ,where a,...
 3.3.32E: Let S be the tetrahedron in with vertices at the vectors 0, be the ...
 3.3.33E: [M] Test the inverse formula of Theorem 8 for a random 4 × 4 matrix...
 3.3.34E: [M] Test Cramer’s rule for a random 4 × 4 matrix A and a random 4 ×...
 3.3.35E: [M] If your version of MATLAB has the flops command, use it to coun...
Solutions for Chapter 3.3: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 3.3
Get Full SolutionsChapter 3.3 includes 35 full stepbystep solutions. 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. Since 35 problems in chapter 3.3 have been answered, more than 32858 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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

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  ).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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