- Chapter 1.1: Vectors
- Chapter 1.2: Dot Product
- Chapter 1.3: Hyperplanes in Rn
- Chapter 1.4: Systems of Linear Equations and Gaussian Elimination
- Chapter 1.5: The Theory of Linear Systems
- Chapter 1.6: Some Applications
- Chapter 2.1: Matrix Operations
- Chapter 2.2: Linear Transformations: An Introduction
- Chapter 2.3: Inverse Matrices
- Chapter 2.4: Elementary Matrices: Rows Get Equal Time
- Chapter 2.5: The Transpose
- Chapter 3.1: Subspaces of Rn
- Chapter 3.2: The Four Fundamental Subspaces
- Chapter 3.3: Linear Independence and Basis
- Chapter 3.4: Dimension and Its Consequences
- Chapter 3.5: A Graphic Example
- Chapter 3.6: AbstractVector Spaces
- Chapter 4.1: Inconsistent Systems and Projection
- Chapter 4.2: Orthogonal Bases
- Chapter 4.3: The Matrix of a Linear Transformation and the Change-of-Basis Formula
- Chapter 4.4: Linear Transformations on Abstract Vector Spaces
- Chapter 5.1: Properties of Determinants
- Chapter 5.2: Cofactors and Cramers Rule
- Chapter 5.3: Signed Area in R2 and SignedVolume in R3
- Chapter 6.1: The Characteristic Polynomial
- Chapter 6.2: Diagonalizability
- Chapter 6.3: Applications
- Chapter 6.4: The Spectral Theorem
- Chapter 7.1: Complex Eigenvalues and Jordan Canonical Form
- Chapter 7.2: Computer Graphics and Geometry
- Chapter 7.3: Matrix Exponentials and Differential Equations
Linear Algebra: A Geometric Approach 2nd Edition - Solutions by Chapter
Full solutions for Linear Algebra: A Geometric Approach | 2nd Edition
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 n-l - An).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
A sequence of steps intended to approach the desired solution.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
= Xl (column 1) + ... + xn(column n) = combination of columns.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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).
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).