 Chapter 1: Linear Equations and Matrices
 Chapter 1.1: Systems of linear Equations
 Chapter 1.2: Matrices
 Chapter 1.3: Matrix Multiplication
 Chapter 1.4: Algebraic Properties of Matrix Operations
 Chapter 1.5: Special Types of Matrices and Partitioned Matrices
 Chapter 1.6: Matrix Transformations
 Chapter 1.7: Computer Graphics (Optional)
 Chapter 1.8: Correlation Coefficient (Optional)
 Chapter 2: Solving linear Systems
 Chapter 2.1: Echelon Form of a Matrix
 Chapter 2.2: Solving Linear Systems
 Chapter 2.3: Elementary Matrices; Finding AI
 Chapter 2.4: Equivalent Matrices
 Chapter 2.5: LUFactorization (Optional)
 Chapter 3: Compute IAI for each of the followin g:
 Chapter 3.1: Definition
 Chapter 3.2: Properties of Determinants
 Chapter 3.3: Cofactor Expansion
 Chapter 3.4: Inverse of a Matrix
 Chapter 3.5: Other Applicotions of Determinants
 Chapter 3.6: Determinants from a Computational Pointof View
 Chapter 4: Real Vector Spaces
 Chapter 4.1: Vectors in the Plane and in 3Spoce
 Chapter 4.2: Vector Spaces
 Chapter 4.3: Subspaces
 Chapter 4.4: Span
 Chapter 4.5: linear Independence
 Chapter 4.6: Basis and Dimension
 Chapter 4.7: Homogeneous Systems
 Chapter 4.8: Coordinates and Isomorphisms
 Chapter 4.9: Coordinates and Isomorphisms
 Chapter 5: looe, Pmd,cI Space,
 Chapter 5.1: length and Diredion in R2 and R3
 Chapter 5.2: Cross Product in R3 (Optional)
 Chapter 5.3: Inner Product Spaces
 Chapter 5.4: GramSchmidt Process
 Chapter 5.5: Orthogonal Complements
 Chapter 5.6: leasl Squares (Optional)
 Chapter 6: Li near Transformations and Matrices
 Chapter 6.1: Definition and Examples
 Chapter 6.2: Kernel and Range of a linear Transformation
 Chapter 6.3: Matrix of a linear Transformation
 Chapter 6.4: Matrix of a linear Transformation
 Chapter 6.5: Similarity
 Chapter 6.6: Introduction to Homogeneous Coordinates (Optional)
 Chapter 7: Eigenvalues and Eigenvectors
 Chapter 7.1: Eigenvalues and Eigenvectors
 Chapter 7.2: Diagonalization and Similar Matrices
 Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
 Chapter 8.2: Spectral Decomposition and Singular Value Decomposition
 Chapter 8.3: Dominanl Eigenvalue and Principal Component Analysis
 Chapter 8.4: Differential Equations
 Chapter 8.5: Dynamical Systems
 Chapter 8.6: Real Quadratic Forms
 Chapter 8.7: Conic Sections
 Chapter 8.8: Quadric Surfaces
Elementary Linear Algebra with Applications 9th Edition  Solutions by Chapter
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Elementary Linear Algebra with Applications  9th Edition  Solutions by Chapter
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Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

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.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.

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