- 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
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.