- 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 Gram--Schmidt 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
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
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
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
A sequence of steps intended to approach the desired solution.
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.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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