 6.3.1: Label the following statements as true or false. Assume that the un...
 6.3.2: Lor each of the following inner product spaces V (over F) and linea...
 6.3.3: For each of the following inner product spaces V and linear operato...
 6.3.4: Complete the proof of Theorem 6.11.
 6.3.5: (a) Complete the proof of the corollary to Theorem 6.11 by using Th...
 6.3.6: Let T be a linear operator on an inner product space V. Let lb = T ...
 6.3.7: Let T be a linear operator on an inner product space V. Let lb = T ...
 6.3.8: Let V be a finitedimensional inner product space, and let T be a l...
 6.3.9: Prove that if V = W W^ and T is the projection on W along W. then ...
 6.3.10: Let T be a linear operator on an inner product space V. Prove that ...
 6.3.11: For a linear operator T on an inner product space V. prove that T*T...
 6.3.12: Let V be an inner product space, and let T be a linear operator on ...
 6.3.13: Let T be a linear operator on a finitedimensional inner product sp...
 6.3.14: Let V be an inner product space, and let, y. z G V'. Define T: V * ...
 6.3.15: The following definition is used in Exercises 1517 and is an exten...
 6.3.16: The following definition is used in Exercises 1517 and is an exten...
 6.3.17: The following definition is used in Exercises 1517 and is an exten...
 6.3.18: Let A be an n x n matrix. Prove that det(.4*) = det(A).
 6.3.19: Suppose that A is an rnxn matrix in which no two columns are identi...
 6.3.20: For each of the sets of data that follows, use the least squares ap...
 6.3.21: In physics. Hooka's law states that (within certain limits) there i...
 6.3.22: Find the minimal solution to each of the following systems of linea...
 6.3.23: Consider the problem of finding the least squares line y = ct + d c...
 6.3.24: Let V and {ei, e 2 ,...} be defined as in Exercise 23 of Section 6....
Solutions for Chapter 6.3: The Adjoint of a Linear Operator
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 6.3: The Adjoint of a Linear Operator
Get Full SolutionsLinear Algebra was written by and is associated to the ISBN: 9780130084514. Since 24 problems in chapter 6.3: The Adjoint of a Linear Operator have been answered, more than 10981 students have viewed full stepbystep solutions from this chapter. Chapter 6.3: The Adjoint of a Linear Operator includes 24 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4.

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).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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).

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

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.