- Chapter 1: Systems of Linear Equations
- Chapter 1-3: Cumulative Test
- 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: Markov Chains
- Chapter 2.6: More Applications of Matrix Operations
- Chapter 3: Determinants
- Chapter 3.1: The Determinant of a Matrix
- Chapter 3.2: Determinants and Elementary Operations
- Chapter 3.3: Properties of Determinants
- Chapter 3.4: Applications of Determinants
- Chapter 4: Vector Spaces
- Chapter 4-5: Cumulative Test
- 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 6-7: Cumulative Test
- 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
Elementary Linear Algebra 8th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra | 8th Edition
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).
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.