 3.5.1: In Exercises 12, let and . Compute the indicated vectors.
 3.5.2: In Exercises 12, let and . Compute the indicated vectors.
 3.5.3: In Exercises 36, use the cross product to find a vector that is ort...
 3.5.4: In Exercises 36, use the cross product to find a vector that is ort...
 3.5.5: In Exercises 36, use the cross product to find a vector that is ort...
 3.5.6: In Exercises 36, use the cross product to find a vector that is ort...
 3.5.7: In Exercises 710, find the area of the parallelogram determined by ...
 3.5.8: In Exercises 710, find the area of the parallelogram determined by ...
 3.5.9: In Exercises 710, find the area of the parallelogram determined by ...
 3.5.10: In Exercises 710, find the area of the parallelogram determined by ...
 3.5.11: In Exercises 1112, find the area of the parallelogram with the give...
 3.5.12: In Exercises 1112, find the area of the parallelogram with the give...
 3.5.13: In Exercises 1314, find the area of the triangle with the given ver...
 3.5.14: In Exercises 1314, find the area of the triangle with the given ver...
 3.5.15: In Exercises 1516, find the area of the triangle in 3space that ha...
 3.5.16: In Exercises 1516, find the area of the triangle in 3space that ha...
 3.5.17: In Exercises 1718, find the volume of the parallelepiped with sides...
 3.5.18: In Exercises 1718, find the volume of the parallelepiped with sides...
 3.5.19: In Exercises 1920, determine whether u, v, and w lie in the same pl...
 3.5.20: In Exercises 1920, determine whether u, v, and w lie in the same pl...
 3.5.21: In Exercises 2124, compute the scalar triple product .
 3.5.22: In Exercises 2124, compute the scalar triple product .
 3.5.23: In Exercises 2124, compute the scalar triple product .
 3.5.24: In Exercises 2124, compute the scalar triple product .
 3.5.25: In Exercises 2526, suppose that . Find
 3.5.26: In Exercises 2526, suppose that . Find
 3.5.27: (a) Find the area of the triangle having vertices , , and . (b) Use...
 3.5.28: Use the cross product to find the sine of the angle between the vec...
 3.5.29: Simplify .
 3.5.30: Let , , , and . Show that
 3.5.31: Let u, v, and w be nonzero vectors in 3space with the same initial...
 3.5.32: Prove the following identities. (a) (b)
 3.5.33: Prove: If a, b, c, and d lie in the same plane, then .
 3.5.34: Prove: If is the angle between u and v and , then .
 3.5.35: Show that if u, v, and w are vectors in no two of which are colline...
 3.5.36: It is a theorem of solid geometry that the volume of a tetrahedron ...
 3.5.37: Use the result of Exercise 26 to find the volume of the tetrahedron...
 3.5.38: Prove part (d) of Theorem 3.5.1. [Hint: First prove the result in t...
 3.5.39: Prove part (e) of Theorem 3.5.1. [Hint: Apply part (a) of Theorem 3...
 3.5.40: Prove: (a) Prove (b) of Theorem 3.5.2. (b) Prove (c) of Theorem 3.5...
 3.5.a: In parts (a)(f) determine whether the statement is true or false, a...
 3.5.b: In parts (a)(f) determine whether the statement is true or false, a...
 3.5.c: In parts (a)(f) determine whether the statement is true or false, a...
 3.5.d: In parts (a)(f) determine whether the statement is true or false, a...
 3.5.e: In parts (a)(f) determine whether the statement is true or false, a...
 3.5.f: In parts (a)(f) determine whether the statement is true or false, a...
Solutions for Chapter 3.5: Cross Product
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 3.5: Cross Product
Get Full SolutionsChapter 3.5: Cross Product includes 46 full stepbystep solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Since 46 problems in chapter 3.5: Cross Product have been answered, more than 13751 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10.

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

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

lAII = 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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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).

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)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.