 3.2.1P: The linear systems are in echelon form. Solve by back substitution.
 3.2.2P: The linear systems are in echelon form. Solve by back substitution.
 3.2.3P: The linear systems are in echelon form. Solve by back substitution.
 3.2.4P: The linear systems are in echelon form. Solve by back substitution.
 3.2.5P: The linear systems are in echelon form. Solve by back substitution.
 3.2.6P: The linear systems are in echelon form. Solve by back substitution.
 3.2.7P: The linear systems are in echelon form. Solve by back substitution.
 3.2.8P: The linear systems are in echelon form. Solve by back substitution.
 3.2.9P: The linear systems are in echelon form. Solve by back substitution.
 3.2.10P: The linear systems are in echelon form. Solve by back substitution.
 3.2.11P: In Problem, use elementary row operations to transform each augment...
 3.2.12P: In Problem, use elementary row operations to transform each augment...
 3.2.13P: In Problem, use elementary row operations to transform each augment...
 3.2.14P: In Problem, use elementary row operations to transform each augment...
 3.2.15P: In Problem, use elementary row operations to transform each augment...
 3.2.16P: In Problem, use elementary row operations to transform each augment...
 3.2.17P: In Problem, use elementary row operations to transform each augment...
 3.2.18P: In Problem, use elementary row operations to transform each augment...
 3.2.19P: In Problem, use elementary row operations to transform each augment...
 3.2.20P: In Problem, use elementary row operations to transform each augment...
 3.2.21P: In Problem, use elementary row operations to transform each augment...
 3.2.22P: In Problem, use elementary row operations to transform each augment...
 3.2.23P: In Problem, determine for what values of k each system has (a) a un...
 3.2.24P: In Problem, determine for what values of k each system has (a) a un...
 3.2.25P: In Problem, determine for what values of k each system has (a) a un...
 3.2.26P: In Problem, determine for what values of k each system has (a) a un...
 3.2.27P: In Problem, determine for what values of k each system has (a) a un...
 3.2.28P: Under what condition on the constants a, b, and c does the system2x...
 3.2.29P: This problem deals with the reversibility of elementary row operati...
 3.2.30P: This problem outlines a proof that two linear systems LS1 and LS2 a...
Solutions for Chapter 3.2: Differential Equations and Linear Algebra 3rd Edition
Full solutions for Differential Equations and Linear Algebra  3rd Edition
ISBN: 9780136054252
Solutions for Chapter 3.2
Get Full SolutionsSince 30 problems in chapter 3.2 have been answered, more than 11471 students have viewed full stepbystep solutions from this chapter. Differential Equations and Linear Algebra was written by and is associated to the ISBN: 9780136054252. This textbook survival guide was created for the textbook: Differential Equations and Linear Algebra, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.2 includes 30 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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].

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

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)ยท(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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.