 2.3.1: For each of the following, compute (i) det(A), (ii) adj A, and (iii...
 2.3.2: Use Cramers rule to solve each of the following systems: (a) x1 + 2...
 2.3.3: Given A = 1 2 1 0 4 3 1 2 2 determine the (2, 3) entry of A1 by com...
 2.3.4: Let A be the matrix in Exercise 3. Compute the third column of A1 b...
 2.3.5: Let A = 1 2 3 2 3 4 3 4 5 (a) Compute the determinant of A. Is A no...
 2.3.6: If A is singular, what can you say about the product A adj A?
 2.3.7: Let Bj denote the matrix obtained by replacing the j th column of t...
 2.3.8: Let A be a nonsingular n n matrix with n > 1. Show that det(adj A) ...
 2.3.9: Let A be a 4 4 matrix. If adj A = 2 0 0 0 0 2 1 0 0 4 3 2 0 2 1 2 (...
 2.3.10: Show that if A is nonsingular, then adj A is nonsingular and (adj A...
 2.3.11: Show that if A is singular, then adj A is also singular. 1
 2.3.12: Show that if det(A) = 1, then adj(adj A) = A 1
 2.3.13: Suppose that Q is a matrix with the property Q1 = QT . Show that qi...
 2.3.14: In coding a message, a blank space was represented by 0, an A by 1,...
 2.3.15: Let x, y, and z be vectors in R3. Show each of the following: (a) x...
 2.3.16: Let x and y be vectors in R3 and define the skewsymmetric matrix Ax...
Solutions for Chapter 2.3: Additional Topics and Applications
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 2.3: Additional Topics and Applications
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Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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].

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).