- 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.
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.
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.
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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!
Solvable system Ax = b.
The right side b is in the column space of A.
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·