 8.2.1: In Exercises 12, determine whether the linear transformation is one...
 8.2.2: In Exercises 12, determine whether the linear transformation is one...
 8.2.3: In Exercises 34, determine whether multiplication by A is onetoon...
 8.2.4: In Exercises 34, determine whether multiplication by A is onetoon...
 8.2.5: Use the given information to determine whether the linear transform...
 8.2.6: Use the given information to determine whether the linear operator ...
 8.2.7: Show that the linear transformation T :P2 R2 defined by T(p(x)) = (...
 8.2.8: Show that the linear transformation T :P2 P2 defined by T(p(x)) = p...
 8.2.9: Let a be a fixed vector in R3. Does the formula T(v) = a v define a...
 8.2.10: Let E be a fixed 2 2 elementary matrix. Does the formula T(A) = EA ...
 8.2.11: In Exercises 1112, compute (T2 T1)(x, y). T1(x, y) = (2x, 3y), T2(x...
 8.2.12: In Exercises 1112, compute (T2 T1)(x, y). T1(x, y) = (2x, 3y, x + y...
 8.2.13: In Exercises 1314, compute (T3 T2 T1)(x, y). T1(x, y) = (2y, 3x, x ...
 8.2.14: In Exercises 1314, compute (T3 T2 T1)(x, y). T1(x, y) = (x + y, y, ...
 8.2.15: Let T1: M22 R and T2: M22 M22 be the linear transformations given b...
 8.2.16: Rework Exercise 15 given that T1:M22 M22 and T2:M22 M22 are the lin...
 8.2.17: Suppose that the linear transformations T1: P2 P2 and T2: P2 P3 are...
 8.2.18: Let T1: Pn Pn and T2: Pn Pn be the linear operators given by T1(p(x...
 8.2.19: Let T : P1 R2 be the function defined by the formula T(p(x)) = (p(0...
 8.2.20: In each part, determine whether the linear operator T : Rn Rn is on...
 8.2.21: Let T : Rn Rn be the linear operator defined by the formula T(x1, x...
 8.2.22: Let T1: R2 R2 and T2: R2 R2 be the linear operators given by the fo...
 8.2.23: Let T1: P2 P3 and T2: P3 P3 be the linear transformations given by ...
 8.2.24: Let TA: R3 R3, TB: R3 R3, and TC: R3 R3 be the reflections about th...
 8.2.25: Let T1: V V be the dilation T1(v) = 4v. Find a linear operator T2: ...
 8.2.26: Let T1:M22 P1 and T2:P1 R3 be the linear transformations given by T...
 8.2.27: Let T : R3 R3 be the orthogonal projection of R3 onto the xyplane....
 8.2.28: (Calculus required) Let V be the vector space C1[0, 1] and let T : ...
 8.2.29: (Calculus required) Let V be the vector space C1[0, 1] and let T : ...
 8.2.30: (Calculus required) Let D(f) = f (x) and J (f) = x 0 f(t) dt be the...
 8.2.31: (Calculus required) Let J :P1 R be the integration transformation J...
 8.2.32: (Calculus required) Let D: Pn Pn1 be the differentiation transforma...
 8.2.33: Prove: If T : V W is a onetoone linear transformation, then T 1: ...
 8.2.34: Use the definition of T3 T2 T1 given by Formula (3) to prove that (...
 8.2.35: Let q0(x) be a fixed polynomial of degree m, and define a function ...
 8.2.36: Prove: If there exists an onto linear transformation T : V W then d...
 8.2.37: Prove: If V and W are finitedimensional vector spaces such that di...
 8.2.TF: TF. In parts (a)(f) determine whether the statement is true or fals...
Solutions for Chapter 8.2: Compositions and InverseTransformations
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 8.2: Compositions and InverseTransformations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 8.2: Compositions and InverseTransformations have been answered, more than 15640 students have viewed full stepbystep solutions from this chapter. Elementary 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 8.2: Compositions and InverseTransformations includes 38 full stepbystep solutions.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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 •

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.