 Chapter 1.1: Systems of Linear Equations
 Chapter 1.10: Systems of Linear Equations
 Chapter 1.2: Row Reduction and Echelon Forms
 Chapter 1.3: Vector Equations
 Chapter 1.4: The Matrix Equation
 Chapter 1.5: Solution Sets of Linear Systems
 Chapter 1.6: Applications of Linear Systems
 Chapter 1.7: Linear Independence
 Chapter 1.8: Introduction to Linear Transformations
 Chapter 1.9: The Matrix of a Linear Transformation
 Chapter 1.SE:
 Chapter 2.1: Matrix Operations
 Chapter 2.2: The Inverse of a Matrix
 Chapter 2.3: Characterizations of Invertible Matrices
 Chapter 2.4: Partitioned Matrices
 Chapter 2.5: Matrix Factorizations
 Chapter 2.6: The Leontief Input–Output Model
 Chapter 2.7: Applications to Computer Graphics
 Chapter 2.8: Subspaces of Rn
 Chapter 2.9: Dimension and Rank
 Chapter 2.SE:
 Chapter 3.1: Introduction to Determinants
 Chapter 3.2: Properties of Determinants
 Chapter 3.3: Cramer’s Rule, Volume, and Linear Transformations
 Chapter 3.SE:
 Chapter 4.1: Vector Spaces and Subspaces
 Chapter 4.2: Null Spaces, Column Spaces, and Linear Transformations
 Chapter 4.3: Linearly Independent Sets; Bases
 Chapter 4.4: Coordinate Systems
 Chapter 4.5: The Dimension of a Vector Space
 Chapter 4.6: Rank
 Chapter 4.7: Change of Basis
 Chapter 4.8: Applications to Difference Equations
 Chapter 4.9: Applications to Markov Chains
 Chapter 4.SE:
 Chapter 5.1: Eigenvectors and Eigenvalues
 Chapter 5.2: The Characteristic Equation
 Chapter 5.3: Diagonalization
 Chapter 5.4: Eigenvectors and Linear Transformations
 Chapter 5.5: Complex Eigenvalues
 Chapter 5.6: Discrete Dynamical Systems
 Chapter 5.7: Applications to Differential Equations
 Chapter 5.8: Iterative Estimates for Eigenvalues
 Chapter 5.SE:
 Chapter 6.1: Inner Product, Length, and Orthogonality
 Chapter 6.2: Orthogonal Sets
 Chapter 6.3: Orthogonal Projections
 Chapter 6.4: The Gram–Schmidt Process
 Chapter 6.5: LeastSquares Problems
 Chapter 6.6: Applications to Linear Models
 Chapter 6.7: Inner Product Spaces
 Chapter 6.8: Applications of Inner Product Spaces
 Chapter 6.SE:
 Chapter 7.1: Diagonalization of Symmetric Matrices
 Chapter 7.2: Quadratic Forms
 Chapter 7.3: Constrained Optimization
 Chapter 7.4: The Singular Value Decomposition
 Chapter 7.5: Applications to Image Processing and Statistics
 Chapter 7.SE:
 Chapter 8.1: Affine Combinations
 Chapter 8.2: Affine Independence
 Chapter 8.3: Convex Combinations
 Chapter 8.4: Hyperplane
 Chapter 8.5: Polytopes
 Chapter 8.6: Curves and Surfaces
Linear Algebra and Its Applications 4th Edition  Solutions by Chapter
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Linear Algebra and Its Applications  4th Edition  Solutions by Chapter
Get Full SolutionsThe full stepbystep solution to problem in Linear Algebra and Its Applications were answered by , our top Math solution expert on 08/10/17, 10:08AM. Since problems from 65 chapters in Linear Algebra and Its Applications have been answered, more than 150787 students have viewed full stepbystep answer. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. This expansive textbook survival guide covers the following chapters: 65.

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).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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.

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

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.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)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.