- 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
Tv = Av + Vo = linear transformation plus shift.
Column space C (A) =
space of all combinations of the columns of A.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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.
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 transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
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 ().
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
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