- Chapter 1: Systems of Linear Equations
- Chapter 1.1: Introduction to Systems of Linear Equations
- Chapter 1.2: Gaussian Elimination and Gauss-Jordan Elimination
- Chapter 1.3: Applications of Systems of Linear Equations
- Chapter 2: Matrices
- Chapter 2.1: Operations with Matrices
- Chapter 2.2: Properties of Matrix Operations
- Chapter 2.3: The Inverse of a Matrix
- Chapter 2.4: Elementary Matrices
- Chapter 2.5: Applications of Matrix Operations
- Chapter 3: Determinants
- Chapter 3.1: The Determinant of a Matrix
- Chapter 3.2: Evaluation of a Determinant Using Elementary Operations
- Chapter 3.3: Properties of Determinants
- Chapter 3.4: Introduction to Eigenvalues
- Chapter 3.5: Applications of Determinants
- Chapter 4: Vector Spaces
- Chapter 4.1: Vectors in Rn
- Chapter 4.2: Vector Spaces
- Chapter 4.3: Subspaces of Vector Spaces
- Chapter 4.4: Spanning Sets and Linear Independence
- Chapter 4.5: Basis and Dimension
- Chapter 4.6: Rank of a Matrix and Systems of Linear Equations
- Chapter 4.7: Coordinates and Change of Basis
- Chapter 4.8: Applications of Vector Spaces
- Chapter 5: Inner Product Spaces
- Chapter 5.1: Length and Dot Product in Rn
- Chapter 5.2: Inner Product Spaces
- Chapter 5.3: Orthonormal Bases: Gram-Schmidt Process
- Chapter 5.4: Mathematical Models and Least Squares Analysis
- Chapter 5.5: Applications of Inner Product Spaces
- Chapter 6: Linear Transformations
- Chapter 6.1: Introduction to Linear Transformations
- Chapter 6.2: The Kernel and Range of a Linear Transformation
- Chapter 6.3: Matrices for Linear Transformations
- Chapter 6.4: Transition Matrices and Similarity
- Chapter 6.5: Applications of Linear Transformations
- Chapter 7: Eigenvalues and Eigenvectors
- Chapter 7.1: Eigenvalues and Eigenvectors
- Chapter 7.2: Diagonalization
- Chapter 7.3: Symmetric Matrices and Orthogonal Diagonalization
- Chapter 7.4: Applications of Eigenvalues and Eigenvectors
- Chapter Appendix: Mathematical Induction and Other Forms of Proofs
Elementary Linear Algebra 6th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra | 6th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Outer product uv T
= column times row = rank one matrix.
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).
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.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.