- Chapter 6.1: If L: V -+ IV is a linear transformation. then for VI and \2 in V. ...
- Chapter 6.2: If L: R! ....... R 2 is a linear transformation ddined by L ([:::])...
- Chapter 6.3: Let L: V -;. IV be a linear transformation. If VI and V2 are in ker...
- Chapter 6.4: If L: V -;. IV is a linear transformation. then for any veclor w in...
- Chapter 6.5: If L : N4 -+ R' is a line;lr transfonnation. then it is possible th...
- Chapter 6.6: A line;lr transfonnation is invertible if and only if it is onto an...
- Chapter 6.7: Similar matrices represent the same linear transformation with resp...
- Chapter 6.8: If a linear transfonnation L : V -;. IV is onto. then the image of ...
- Chapter 6.9: If a linear transfonllation L: Rl -+ H) is onto, then L is invertible.
- Chapter 6.10: If L : V -+ IV is a linear transformation. then the image of a line...
- Chapter 6.11: Similar matrices have the same detenninanl.
- Chapter 6.12: The determinant of 3 x 3 matrices defines a linear transfonnation f...
Solutions for Chapter Chapter 6: Li near Transformations and Matrices
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).