- 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
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Dimension of vector space
dim(V) = number of vectors in any basis for V.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
A directed graph that has constants Cl, ... , Cm associated with the edges.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
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.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).
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