- 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 A-I
- Chapter 2.4: Equivalent Matrices
- Chapter 2.5: LU-Factorization (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 3-Spoce
- 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: Gram-Schmidt 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
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
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, off-diagonal 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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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. A-I 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 A-I 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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).