- Chapter 1: Linear Equations
- Chapter 1.1: Introduction to Linear Systems
- Chapter 1.2: Matrices, Vectors, and Gauss-Jordan Elimination
- Chapter 1.3: On the Solutions of Linear Systems; Matrix Algebra
- Chapter 2: Linear Transformations
- Chapter 2.1: Introduction to Linear Transformations and Their Inverses
- Chapter 2.2: Linear Transformations in Geometry
- Chapter 2.3: Matrix Products
- Chapter 2.4: The Inverse of a Linear Transformation
- Chapter 3.1: Image and Kernel of a Linear Transformation
- Chapter 3.2: Subspaces of R"; Bases and Linear Independence
- Chapter 3.3: The Dimension of a Subspace of R"
- Chapter 3.4: Coordinates
- Chapter 4: Linear Spaces
- Chapter 4.1: Introduction to Linear Spaces
- Chapter 4.2: Linear Transformations and Isomorphisms
- Chapter 4.3: Th e Matrix of a Linear Transformation
- Chapter 5: Orthogonality and Least Squares
- Chapter 5.1: Orthogonal Projections and Orthonormal Bases
- Chapter 5.2: Gram-Schmidt Process and QR Factorization
- Chapter 5.3: Orthogonal Transformations and Orthogonal Matrices
- Chapter 5.4: Least Squares and Data Fitting
- Chapter 5.5: Inner Product Spaces
- Chapter 6: Determinants
- Chapter 6.1: Introduction to Determinants
- Chapter 6.2: Properties of the Determinant
- Chapter 6.3: Geometrical Interpretations of the Determinant; Cramers Rule
- Chapter 7: Eigenvalues and Eigenvectors
- Chapter 7.1: Dynamical Systems and Eigenvectors: An Introductory Example
- Chapter 7.2: Finding the Eigenvalues of a Matrix
- Chapter 7.3: Finding the Eigenvectors of a Matrix
- Chapter 7.4: Diagonalization
- Chapter 7.5: Complex Eigenvalues
- Chapter 7.6: Stability
- Chapter 8: Symmetric Matrices and Quadratic Forms
- Chapter 8.1: Symmetric Matrices
- Chapter 8.2: Quadratic Forms
- Chapter 8.3: Singular Values
- Chapter 9.1: An Introduction to Continuous Dynamical Systems
- Chapter 9.2: The Complex Case: Eulers Formula
- Chapter 9.3: Linear Differential Operators and Linear Differential Equations
Linear Algebra with Applications 4th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 4th Edition
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).
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
peA) = det(A - AI) has peA) = zero matrix.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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!
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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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