 Chapter 1.1: Vectors and Linear Combinations
 Chapter 1.2: Lengths and Dot Products
 Chapter 1.3: Matrices
 Chapter 10.1: Complex Numbers
 Chapter 10.2: Hermitian and Unitary Matrices
 Chapter 10.3: The Fast Fourier Transform
 Chapter 2.1: Solving Linear Equations
 Chapter 2.2: The Idea of Elimination
 Chapter 2.3: Elimination Using Matrices
 Chapter 2.4: Rules for Matrix Operations
 Chapter 2.5: Inverse Matrices
 Chapter 2.6: Elimination = Factorization: A = L U
 Chapter 2.7: Transposes and Permutations
 Chapter 3.1: Spaces of Vectors
 Chapter 3.2: The Nullspace of A: Solving Ax = 0
 Chapter 3.3: The Rank and the Row Reduced Form
 Chapter 3.4: The Complete Solution to Ax = b
 Chapter 3.5: Independence, Basis and Dimension
 Chapter 3.6: Dimensions of the Four Subspaces
 Chapter 4.1: Orthogonality of the Four Subspaces
 Chapter 4.2: Projections
 Chapter 4.3: Least Squares Approximations
 Chapter 4.4: Orthogonal Bases and GramSchmidt
 Chapter 5.1: The Properties of Determinants
 Chapter 5.2: Permutations and Cofactors
 Chapter 5.3: Cramer's Rule, Inverses, and Volumes
 Chapter 6.1: Introduction to Eigenvalues
 Chapter 6.2: Diagonalizing a Matrix
 Chapter 6.3: Applications to Differential Equations
 Chapter 6.4: Symmetric Matrices
 Chapter 6.5: Positive Definite Matrices
 Chapter 6.6: Similar Matrices
 Chapter 6.7: Singular Value Decomposition (SVD)
 Chapter 7.1: The Idea of a Linear Transformation
 Chapter 7.2: The Matrix of a Linear Transformation
 Chapter 7.3: Diagonalization and the Pseudoinverse
 Chapter 8.1: Matrices in Engineering
 Chapter 8.2: Graphs and Networks
 Chapter 8.3: Markov Matrices, Population, and Economics
 Chapter 8.4: Linear Programming
 Chapter 8.5: Fourier Series: Linear Algebra for Functions
 Chapter 8.6: Linear Algebra for Statistics and Probability
 Chapter 8.7: Computer Graphics
 Chapter 9.1: Gaussian Elimination in Practice
 Chapter 9.2: Norms and Condition Numbers
 Chapter 9.3: Iterative Methods and Preconditioners
Introduction to Linear Algebra 4th Edition  Solutions by Chapter
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Introduction to Linear Algebra  4th Edition  Solutions by Chapter
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

Graph G.
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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

Outer product uv T
= column times row = rank one matrix.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.