 8.1: Let A be an n n matrix, B a nonzero n 1 matrix, and x a vector in R...
 8.2: . Let A = cos sin sin cos (a) Show that A2 = cos 2 sin 2 sin 2 cos ...
 8.3: Devise a method for finding two n n matrices that are not similar. ...
 8.4: Let v1, v2,..., vm be fixed vectors in Rn, and let T : Rn Rm be the...
 8.5: Let {e1, e2, e3, e4} be the standard basis for R4, and let T : R4 R...
 8.6: Suppose that vectors in R3 are denoted by 1 3 matrices, and define ...
 8.7: Let B = {v1, v2, v3, v4} be a basis for a vector space V, and let T...
 8.8: Let V and W be vector spaces, let T , T1, and T2 be linear transfor...
 8.9: Let A and B be similar matrices. Prove: (a) AT and BT are similar. ...
 8.10: (Fredholm Alternative Theorem) Let T : V V be a linear operator on ...
 8.11: Let T : M22 M22 be the linear operator defined by T(X) = 1 1 0 0 X ...
 8.12: Prove: If A and B are similar matrices, and if B and C are also sim...
 8.13: Let L: M22 M22 be the linear operator that is defined by L(M) = MT ...
 8.14: Let B = {u1, u2, u3} and B = {v1, v2, v3} be bases for a vector spa...
 8.15: Let B = {u1, u2, u3} be a basis for a vector space V, and let T : V...
 8.16: Show that the matrices 1 1 1 4 and 2 1 1 3 are similar but that 3 1...
 8.17: Suppose that T : V V is a linear operator, and B is a basis for V f...
 8.18: Let T : V V be a linear operator. Prove that T is onetoone if and...
 8.19: (Calculus required) (a) Show that if f = f(x) is twice differentiab...
 8.20: Let T : P2 R3 be the function defined by the formula T(p(x)) = p(1)...
 8.21: Let x1, x2, and x3 be distinct real numbers such that x1 < x2 < x3 ...
 8.22: (Calculus required) Let p(x) and q(x) be continuous functions, and ...
 8.23: (Calculus required) Let D: Pn Pn be the differentiation operator D(...
 8.24: (Calculus required) It can be shown that for any real number c, the...
 8.25: (Calculus required) Let J : Pn Pn+1 be the integration transformati...
Solutions for Chapter 8: General Linear Transformations
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 8: General Linear Transformations
Get Full SolutionsElementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Chapter 8: General Linear Transformations includes 25 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 25 problems in chapter 8: General Linear Transformations have been answered, more than 16034 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Outer product uv T
= column times row = rank one matrix.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as 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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).