- 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
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.