Section 6.3 2 Solution Linear Algebra Lays
Solutions for Chapter 6.3
Textbook: Linear Algebra and Its Applications
Edition: 5
Author: David C. Lay; Steven R. Lay; Judi J. McDonald
ISBN: 9780321982384
Chapter 6.3 includes 26 full step-by-step 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: 9780321982384. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Since 26 problems in chapter 6.3 have been answered, more than 118093 students have viewed full step-by-step solutions from this chapter.
- 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).
- Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
- Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
- 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.
- Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
- Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
- Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
- 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.
- Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
- Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
- Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
- Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
- 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.
- Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
- Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.
- Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
- Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
- Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
- Unitary matrix UH = U T = U-I.
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.
-
About us
Team
Careers
Blog
-
Schools
Subjects
Textbook Survival Guides
-
Elite Notetakers
Referral Program
Campus Marketing Coordinators
Scholarships
-
Contact
FAQ
Sitemap
Section 6.3 2 Solution Linear Algebra Lays
Source: https://studysoup.com/tsg/math/49/linear-algebra-and-its-applications/chapter/747/6-3