- Chapter 1: Matrices and Systems of Equations
- Chapter 1.1: Systems of Linear Equations
- Chapter 1.2: Row Echelon Form
- Chapter 1.3: Matrix Arithmetic
- Chapter 1.4: Matrix Algebra
- Chapter 1.5: Elementary Matrices
- Chapter 1.6: Partitioned Matrices
- Chapter 2: Determinants
- Chapter 2.1: The Determinant of a Matrix
- Chapter 2.2: Properties of Determinants
- Chapter 2.3: Additional Topics and Applications
- Chapter 3: Vector Spaces
- Chapter 3.1: Definition and Examples
- Chapter 3.2: Subspaces
- Chapter 3.3: Linear Independence
- Chapter 3.4: Basis and Dimension
- Chapter 3.5: Change of Basis
- Chapter 3.6: Row Space and Column Space
- Chapter 4: Linear Transformations
- Chapter 4.1: Definition and Examples
- Chapter 4.2: Matrix Representations of Linear Transformations
- Chapter 4.3: Similarity
- Chapter 5: Orthogonality
- Chapter 5.1: The Scalar Product in Rn
- Chapter 5.2: Orthogonal Subspaces
- Chapter 5.3: Least Squares Problems
- Chapter 5.4: Inner Product Spaces
- Chapter 5.5: Orthonormal Sets
- Chapter 5.6: The GramSchmidt Orthogonalization Process
- Chapter 5.7: Orthogonal Polynomials
- Chapter 6: Eigenvalues
- Chapter 6.1: Eigenvalues and Eigenvectors
- Chapter 6.2: Systems of Linear Differential Equations
- Chapter 6.3: Diagonalization
- Chapter 6.4: Hermitian Matrices
- Chapter 6.5: The Singular Value Decomposition
- Chapter 6.6: Quadratic Forms
- Chapter 6.7: Positive Definite Matrices
- Chapter 6.8: Nonnegative Matrices
- Chapter 7: Numerical Linear Algebra
- Chapter 7.1: Floating-Point Numbers
- Chapter 7.2: Gaussian Elimination
- Chapter 7.3: Pivoting Strategies
- Chapter 7.4: Matrix Norms and Condition Numbers
- Chapter 7.5: Orthogonal Transformations
- Chapter 7.6: The Eigenvalue Problem
- Chapter 7.7: Least Squares Problems
Linear Algebra with Applications 8th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 8th Edition
Column space C (A) =
space of all combinations of the columns of A.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def 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!
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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