- Chapter 1.1: Vectors and Linear Combinations
- Chapter 1.2: Lengths and Dot Products
- Chapter 1.3: Matrices
- Chapter 10.1: Complex Numbers
- Chapter 10.2: Hermitian and Unitary Matrices
- Chapter 10.3: The Fast Fourier Transform
- Chapter 2.1: Solving Linear Equations
- Chapter 2.2: The Idea of Elimination
- Chapter 2.3: Elimination Using Matrices
- Chapter 2.4: Rules for Matrix Operations
- Chapter 2.5: Inverse Matrices
- Chapter 2.6: Elimination = Factorization: A = L U
- Chapter 2.7: Transposes and Permutations
- Chapter 3.1: Spaces of Vectors
- Chapter 3.2: The Nullspace of A: Solving Ax = 0
- Chapter 3.3: The Rank and the Row Reduced Form
- Chapter 3.4: The Complete Solution to Ax = b
- Chapter 3.5: Independence, Basis and Dimension
- Chapter 3.6: Dimensions of the Four Subspaces
- Chapter 4.1: Orthogonality of the Four Subspaces
- Chapter 4.2: Projections
- Chapter 4.3: Least Squares Approximations
- Chapter 4.4: Orthogonal Bases and Gram-Schmidt
- Chapter 5.1: The Properties of Determinants
- Chapter 5.2: Permutations and Cofactors
- Chapter 5.3: Cramer's Rule, Inverses, and Volumes
- Chapter 6.1: Introduction to Eigenvalues
- Chapter 6.2: Diagonalizing a Matrix
- Chapter 6.3: Applications to Differential Equations
- Chapter 6.4: Symmetric Matrices
- Chapter 6.5: Positive Definite Matrices
- Chapter 6.6: Similar Matrices
- Chapter 6.7: Singular Value Decomposition (SVD)
- Chapter 7.1: The Idea of a Linear Transformation
- Chapter 7.2: The Matrix of a Linear Transformation
- Chapter 7.3: Diagonalization and the Pseudoinverse
- Chapter 8.1: Matrices in Engineering
- Chapter 8.2: Graphs and Networks
- Chapter 8.3: Markov Matrices, Population, and Economics
- Chapter 8.4: Linear Programming
- Chapter 8.5: Fourier Series: Linear Algebra for Functions
- Chapter 8.6: Linear Algebra for Statistics and Probability
- Chapter 8.7: Computer Graphics
- Chapter 9.1: Gaussian Elimination in Practice
- Chapter 9.2: Norms and Condition Numbers
- Chapter 9.3: Iterative Methods and Preconditioners
Introduction to Linear Algebra 4th Edition - Solutions by Chapter
Full solutions for Introduction to Linear Algebra | 4th Edition
ISBN: 9780980232714
Introduction to Linear Algebra | 4th Edition - Solutions by Chapter
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Solutions by Chapter
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Key Math Terms and definitions covered in this textbook
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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).
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Column space C (A) =
space of all combinations of the columns of A.
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Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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Covariance matrix:E.
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.
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Dimension of vector space
dim(V) = number of vectors in any basis for V.
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Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
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Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
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Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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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.
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.