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- 6.4.7: Let Ll PI -+ P2 be the linear transformation defined by L1 (p(t) = ...
- 6.4.8: Let LI. L,. and S be as in Exercise 3. Find the following: (a) (L 1...
- 6.4.9: If [! _: -iJ , .. he "pre~m"'o<' of' h'"'' 0",' ator L: R 3 -+ R3 w...
- 6.4.10: Let L j L,. and L3 be linear transformations of R3 into R, defined ...
- 6.4.11: Find the dimension of the vector space U of all linear transformati...
- 6.4.12: Repeat Exercise II for each of the fo llowing: (a ) V = IV is the v...
- 6.4.13: Let A = [ali ] be a given /II x I! matrix. and let V and IV be give...
- 6.4.14: Let A = [~ ~ ~ 1 Let S be the natural basis for Rl and T be the nat...
- 6.4.15: Let A be as in Exercise 14. Consider the ordered bases S = if'. I. ...
- 6.4.16: Find two linear transformations L j : H' -4 H: and L 2: H2 _ R'such...
- 6.4.17: Find a linear transformation L : R2 -4 K'. L '" f, the identity ope...
- 6.4.18: Find a linear transformation L: H' ........ H'. L '" O. the zero tr...
- 6.4.19: Find a linear transformation L: R 2 -)- H2. L I- I. L '" O ,such th...
- 6.4.20: Let L : R 3 -)- Rl be the linear transformation defined in Exercise...
- 6.4.21: Let L : R 3 ..... R3 be the linear transformation definoo in Exerci...
- 6.4.22: Let L : R' ..... Rl be the invertible linear transfonr.ation repres...
- 6.4.23: Let L: V ..... V be a linear transfonnation represented by a matrix...
- 6.4.24: Let L: P, -+ P, be the invertible linear transfonr.ation represente...
Solutions for Chapter 6.4: Matrix of a linear Transformation
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
Remove row i and column j; multiply the determinant by (-I)i + j •
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.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-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
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = 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.
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.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.