 8.3.1: Find . (a) , (b) , (c) , (d)
 8.3.2: Find . (a) , , (b)
 8.3.3: Let and be the linear transformations given by and . (a) Find , whe...
 8.3.4: Let and be the linear operators given by and . Find and
 8.3.5: Let be the dilation . Find a linear operator such that and
 8.3.6: Suppose that the linear transformations and are given by the formulas
 8.3.7: Let be a fixed polynomial of degree m, and define a function T with...
 8.3.8: Use the definition of given by Formula 3 to prove that (a) is a lin...
 8.3.9: Let be the orthogonal projection of onto the xyplane. Show that .
 8.3.10: In each part, let be multiplication by A. Determine whether T has a...
 8.3.11: In each part, let be multiplication by A. Determine whether T has a...
 8.3.12: In each part, determine whether the linear operator is onetoone; ...
 8.3.13: Let be the linear operator defined by the formula where are constan...
 8.3.14: Let and be the linear operators given by the formulas (a) Show that...
 8.3.15: Let and be the linear transformations given by the formulas (a) Fin...
 8.3.16: Let , , and be the reflections about the xyplane, the plane, and ...
 8.3.17: Let be the function defined by the formula (a) Find . (b) Show that...
 8.3.18: Let be the linear operator given by the formula . Show that T is on...
 8.3.19: Prove: If is a onetoone linear transformation, then is a onetoo...
 8.3.20: In Exercises 2021, determine whether .(a) is the orthogonal project...
 8.3.21: In Exercises 2021, determine whether .(a) is the reflection about t...
 8.3.22: Calculus required) Let be the linear transformations in Examples 11...
 8.3.23: (Calculus required) The Fundamental Theorem of Calculus implies tha...
 8.3.a: In parts (a)(f) determine whether the statement is true or false, a...
 8.3.b: In parts (a)(f) determine whether the statement is true or false, a...
 8.3.c: In parts (a)(f) determine whether the statement is true or false, a...
 8.3.d: In parts (a)(f) determine whether the statement is true or false, a...
 8.3.e: In parts (a)(f) determine whether the statement is true or false, a...
 8.3.f: In parts (a)(f) determine whether the statement is true or false, a...
Solutions for Chapter 8.3: Compositions and Inverse Transformations
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 8.3: Compositions and Inverse Transformations
Get Full SolutionsElementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Chapter 8.3: Compositions and Inverse Transformations includes 29 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Since 29 problems in chapter 8.3: Compositions and Inverse Transformations have been answered, more than 13761 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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 •

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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!

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).