- Chapter 1: Systems of Linear Equations
- Chapter 2: Matrices
- Chapter 3: Determinants
- Chapter 4: Vector Spaces
- Chapter 5: Inner Product Spaces
- Chapter 6: Inner Product Spaces
- Chapter 7: Eigenvalues and Eigenvectors
Elementary Linear Algebra 7th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra | 7th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Remove row i and column j; multiply the determinant by (-I)i + j •
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
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
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Constant down each diagonal = time-invariant (shift-invariant) filter.
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