 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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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Incidence matrix of a directed graph.
The m by n edgenode 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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.