- Chapter 1.1: Systems of linear Equations
- Chapter 1.2: Matrices
- Chapter 1.3: Matrix Multiplication
- Chapter 1.4: Algebraic Properties of Matrix Operations
- Chapter 1.5: Special Types of Matrices and Partitioned Matrices
- Chapter 1.6: Matrix Transformations
- Chapter 1.7: Computer Graphics (Optional)
- Chapter 1.8: Correlation Coefficient (Optional)
- Chapter 2.1: Echelon Form of a Matrix
- Chapter 2.2: Solving Linear Systems
- Chapter 2.3: Elementary Matrices; Finding A-I
- Chapter 2.4: Equivalent Matrices
- Chapter 2.5: LU-Factorization (Optional)
- Chapter 3.1: Definition
- Chapter 3.2: Properties of Determinants
- Chapter 3.3: Cofactor Expansion
- Chapter 3.4: Inverse of a Matrix
- Chapter 3.5: Other Applicotions of Determinants
- Chapter 3.6: Determinants from a Computational Pointof View
- Chapter 4.1: Vectors in the Plane and in 3-Spoce
- Chapter 4.2: Vector Spaces
- Chapter 4.3: Subspaces
- Chapter 4.4: Span
- Chapter 4.5: linear Independence
- Chapter 4.6: Basis and Dimension
- Chapter 4.7: Homogeneous Systems
- Chapter 4.8: Coordinates and Isomorphisms
- Chapter 4.9: Coordinates and Isomorphisms
- Chapter 5.1: length and Diredion in R2 and R3
- Chapter 5.2: Cross Product in R3 (Optional)
- Chapter 5.3: Inner Product Spaces
- Chapter 5.4: Gram-Schmidt Process
- Chapter 5.5: Orthogonal Complements
- Chapter 5.6: leasl Squares (Optional)
- Chapter 6.1: Definition and Examples
- Chapter 6.2: Kernel and Range of a linear Transformation
- Chapter 6.3: Matrix of a linear Transformation
- Chapter 6.4: Matrix of a linear Transformation
- Chapter 6.5: Similarity
- Chapter 6.6: Introduction to Homogeneous Coordinates (Optional)
- Chapter 7.1: Eigenvalues and Eigenvectors
- Chapter 7.2: Diagonalization and Similar Matrices
- Chapter 8.1: Stable Age Distribution in a Population; Markov Processes
- Chapter 8.2: Spectral Decomposition and Singular Value Decomposition
- Chapter 8.3: Dominanl Eigenvalue and Principal Component Analysis
- Chapter 8.4: Differential Equations
- Chapter 8.5: Dynamical Systems
- Chapter 8.6: Real Quadratic Forms
- Chapter 8.7: Conic Sections
- Chapter 8.8: Quadric Surfaces
- Chapter Chapter 1: Linear Equations and Matrices
- Chapter Chapter 2: Solving linear Systems
- Chapter Chapter 3: Compute IAI for each of the followin g:
- Chapter Chapter 4: Real Vector Spaces
- Chapter Chapter 5: looe, Pmd,cI Space,
- Chapter Chapter 6: Li near Transformations and Matrices
- Chapter Chapter 7: Eigenvalues and Eigenvectors
Elementary Linear Algebra with Applications 9th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
= Xl (column 1) + ... + xn(column n) = combination of columns.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Constant down each diagonal = time-invariant (shift-invariant) filter.
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.
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