 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
ISBN: 9780471669593
Solutions for Chapter Chapter 6: Li near Transformations and Matrices
Get Full SolutionsSince 12 problems in chapter Chapter 6: Li near Transformations and Matrices have been answered, more than 9106 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This expansive textbook survival guide covers the following chapters and their solutions. Chapter Chapter 6: Li near Transformations and Matrices includes 12 full stepbystep solutions.

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 IIA1 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 Fn1c 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.

Fundamental Theorem.
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, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Schwarz inequality
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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) 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 = UI.
Orthonormal columns (complex analog of Q).