 Chapter 1.1: Lines and Linear Equations
 Chapter 1.2: Linear Systems and Matrices
 Chapter 1.3: Numerical Solutions
 Chapter 1.4: Applications of Linear Systems
 Chapter 10.1: Inner Products
 Chapter 10.2: The GramSchmidt Process Revisited
 Chapter 10.3: Applications of Inner Products
 Chapter 11.1: Quadratic Forms
 Chapter 11.2: Positive Definite Matrices
 Chapter 11.3: Constrained Optimization
 Chapter 11.4: Complex Vector Spaces
 Chapter 11.5: Hermitian Matrices
 Chapter 2.1: Vectors
 Chapter 2.2: Span
 Chapter 2.3: Linear Independence
 Chapter 3.1: Linear Transformations
 Chapter 3.2: Matrix Algebra
 Chapter 3.3: Inverses
 Chapter 3.4: LU Factorization
 Chapter 3.5: Markov Chains
 Chapter 4.1: Introduction to Subspaces
 Chapter 4.2: Basis and Dimension
 Chapter 4.3: Row and Column Spaces
 Chapter 5.1: The Determinant Function
 Chapter 5.2: Properties of the Determinant
 Chapter 5.3: Applications of the Determinant
 Chapter 6.1: Eigenvalues and Eigenvectors
 Chapter 6.2: Approximation Methods
 Chapter 6.3: Change of Basis
 Chapter 6.4: Diagonalization
 Chapter 6.5: Complex Eigenvalues
 Chapter 6.6: Systems of Differential Equations
 Chapter 7.1: Vector Spaces and Subspaces
 Chapter 7.2: Span and Linear Independence
 Chapter 7.3: Basis and Dimension
 Chapter 8.1: Dot Products and Orthogonal Sets
 Chapter 8.2: Projection and the GramSchmidt Process
 Chapter 8.3: Diagonalizing Symmetric Matrices and QR Factorization
 Chapter 8.4: The Singular Value Decomposition
 Chapter 8.5: Least Squares Regression
 Chapter 9.1: Definition and Properties
 Chapter 9.2: Isomorphisms
 Chapter 9.3: The Matrix of a Linear Transformation
 Chapter 9.4: Similarity
Linear Algebra with Applications 1st Edition  Solutions by Chapter
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Linear Algebra with Applications  1st Edition  Solutions by Chapter
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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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Pascal matrix
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).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.