- 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 Lint!ar Systt!ms
- Chapter 2.3: Elementary Matrices; Finding A -
- Chapter 2.4: Equivalent Matrices
- Chapter 2.5: LV-Factorization (Optional)
- Chapter 3: Determinants
- 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 Applications of Determinants
- Chapter 3.6: Determinants from a Computational Point of View
- Chapter 4: Real Vector Spaces
- Chapter 4.1: Vectors in the Plane and in 3-Space
- 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: Rank of a Matrix
- Chapter 5: Inner Product Spaces
- Chapter 5.1: Length and Direction in R2 and R3
- Chapter 5.2: Cross Product in RJ (Optional)
- Chapter 5.3: Inner Product Spaces
- Chapter 5.4: Gram*-Schmidtt Process
- Chapter 5.5: Orthogonal Complements
- Chapter 5.6: Least 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: Vector Space of Matrices and Vector Space of Linear Transformations (Optional)
- 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 7.3: Diagonalization of Symmetric Matrices
- Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
- Chapter 8.2: Spectral Decomposition and Singular Value Decomposition
- Chapter 8.3: Dominant Eigenvalue and Principal Component Analysis
- Chapter 8.4: Differential Equations
- 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
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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.
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)·(b - Ax) = o.
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.
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.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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