- 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
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
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
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
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].
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
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.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Every v in V is orthogonal to every w in W.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.