 4.4.1E: In Exercises 1–4, find the vector x determined by the given coordin...
 4.4.2E: Find the vector × determined by the given coordinate vector and the...
 4.4.3E: Find the vector × determined by the given coordinate vector and the...
 4.4.4E: Find the vector × determined by the given coordinate vector and the...
 4.4.5E: Find the coordinate vector of × relative to the given basis = {b1,....
 4.4.6E: Find the coordinate vector of × relative to the given basis = {b1,....
 4.4.7E: In Exercises 5–8, find the coordinate vector of x relative to the g...
 4.4.8E: Find the coordinate vector of × relative to the given basis = {b1,....
 4.4.9E: Find the changeofcoordinates matrix from to the standard basis in...
 4.4.10E: Find the changeofcoordinates matrix from to the standard basis in...
 4.4.11E: Use an inverse matrix to find for the given × and .
 4.4.12E: Use an inverse matrix to find for the given × and .
 4.4.13E: The set Find the coordinate vector of relative to .
 4.4.14E: The set Find the coordinate vector of relative to
 4.4.15E: In Exercises 15 and 16, mark each statement True or False. Justify ...
 4.4.16E: In Exercises 15 and 16, mark each statement True or False. Justify ...
 4.4.17E: The vectors but do not form a basis. Find two different ways to exp...
 4.4.18E: Let be a basis for a vector space V. Explain why the Bcoordinate v...
 4.4.19E: Let S be a finite set in a vector space V with the property that ev...
 4.4.20E: Suppose is a linearly dependent spanning set for a vector space V. ...
 4.4.21E: Since the coordinate mapping determined by B is a linear transforma...
 4.4.22E: Produce a description of an n × n matrix A that implements the coor...
 4.4.23E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.24E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.25E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.26E: Exercises 23–26 concern a vector space V, a basis and the coordinat...
 4.4.27E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.28E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.29E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.30E: In Exercises 27–30, use coordinate vectors to test the linear indep...
 4.4.31E: Use coordinate vectors to test whether the following sets of polyno...
 4.4.32E: a. Use coordinate vectors to show that these polynomials form a bas...
 4.4.33E: In Exercises 33 and 34, determine whether the sets of polynomials f...
 4.4.34E: In Exercises 33 and 34, determine whether the sets of polynomials f...
 4.4.35E: Show that x is in H and find the coordinate vector of x, for
 4.4.36E: Show that is a basis forH and x is inH, and find the coordinate ve...
 4.4.37E: [M] Exercises 37 and 38 concern the crystal lattice for titanium, w...
 4.4.38E: [M] Exercises 37 and 38 concern the crystal lattice for titanium, w...
Solutions for Chapter 4.4: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 4.4
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.4 includes 38 full stepbystep solutions. Since 38 problems in chapter 4.4 have been answered, more than 43116 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384.

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).

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

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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.

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

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.