 Chapter 1: Systems of Linear Equations
 Chapter 1.1: Introduction to Systems of Linear Equations
 Chapter 1.2: Gaussian Elimination and GaussJordan Elimination
 Chapter 1.3: Applications of Systems of Linear Equations
 Chapter 2: Matrices
 Chapter 2.1: Operations with Matrices
 Chapter 2.2: Properties of Matrix Operations
 Chapter 2.3: The Inverse of a Matrix
 Chapter 2.4: Elementary Matrices
 Chapter 2.5: Applications of Matrix Operations
 Chapter 3: Determinants
 Chapter 3.1: The Determinant of a Matrix
 Chapter 3.2: Evaluation of a Determinant Using Elementary Operations
 Chapter 3.3: Properties of Determinants
 Chapter 3.4: Introduction to Eigenvalues
 Chapter 3.5: Applications of Determinants
 Chapter 4: Vector Spaces
 Chapter 4.1: Vectors in Rn
 Chapter 4.2: Vector Spaces
 Chapter 4.3: Subspaces of Vector Spaces
 Chapter 4.4: Spanning Sets and Linear Independence
 Chapter 4.5: Basis and Dimension
 Chapter 4.6: Rank of a Matrix and Systems of Linear Equations
 Chapter 4.7: Coordinates and Change of Basis
 Chapter 4.8: Applications of Vector Spaces
 Chapter 5: Inner Product Spaces
 Chapter 5.1: Length and Dot Product in Rn
 Chapter 5.2: Inner Product Spaces
 Chapter 5.3: Orthonormal Bases: GramSchmidt Process
 Chapter 5.4: Mathematical Models and Least Squares Analysis
 Chapter 5.5: Applications of Inner Product Spaces
 Chapter 6: Linear Transformations
 Chapter 6.1: Introduction to Linear Transformations
 Chapter 6.2: The Kernel and Range of a Linear Transformation
 Chapter 6.3: Matrices for Linear Transformations
 Chapter 6.4: Transition Matrices and Similarity
 Chapter 6.5: Applications of Linear Transformations
 Chapter 7: Eigenvalues and Eigenvectors
 Chapter 7.1: Eigenvalues and Eigenvectors
 Chapter 7.2: Diagonalization
 Chapter 7.3: Symmetric Matrices and Orthogonal Diagonalization
 Chapter 7.4: Applications of Eigenvalues and Eigenvectors
 Chapter Appendix: Mathematical Induction and Other Forms of Proofs
Elementary Linear Algebra 6th Edition  Solutions by Chapter
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Elementary Linear Algebra  6th Edition  Solutions by Chapter
Get Full SolutionsElementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. The full stepbystep solution to problem in Elementary Linear Algebra were answered by , our top Math solution expert on 03/13/18, 08:31PM. This expansive textbook survival guide covers the following chapters: 43. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Since problems from 43 chapters in Elementary Linear Algebra have been answered, more than 6575 students have viewed full stepbystep answer.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Column space C (A) =
space of all combinations of the columns of A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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.

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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