 8.4.1: Let T : P2 P3 be the linear transformation defined by T(p(x)) = xp(...
 8.4.2: Let T : P2 P1 be the linear transformation defined by T(a0 + a1x + ...
 8.4.3: Let T : P2 P2 be the linear operator defined by T(a0 + a1x + a2x2 )...
 8.4.4: Let T : R2 R2 be the linear operator defined by T x1 x2 ! = x1 x2 x...
 8.4.5: Let T : R2 R3 be defined by T x1 x2 ! = x1 + 2x2 x1 0 (a) Find the ...
 8.4.6: Let T : R3 R3 be the linear operator defined by T(x1, x2, x3) = (x1...
 8.4.7: Let T : P2 P2 be the linear operator defined by T(p(x)) = p(2x + 1)...
 8.4.8: Let T : P2 P3 be the linear transformation defined by T(p(x)) = xp(...
 8.4.9: Let v1 = 1 3 and v2 = 1 4 , and let A = 1 3 2 5 be the matrix for T...
 8.4.10: Let A = 3 210 1621 3071 be the matrix for T : R4 R3 relative to the...
 8.4.11: Let A = 1 3 1 205 6 2 4 be the matrix for T : P2 P2 with respect to...
 8.4.12: Let T1: P1 P2 be the linear transformation defined by T1(p(x)) = xp...
 8.4.13: Let T1: P1 P2 be the linear transformation defined by T1(c0 + c1x) ...
 8.4.14: Let B = {v1, v2, v3, v4} be a basis for a vector space V. Find the ...
 8.4.15: . Let T : P2 M22 be the linear transformation defined by T (p) = p(...
 8.4.16: Let T : M22 R2 be the linear transformation given by T a b c d = a ...
 8.4.17: (Calculus required) Let D: P2 P2 be the differentiation operator D(...
 8.4.18: (Calculus required) Let D: P2 P2 be the differentiation operator D(...
 8.4.19: (Calculus required) Let V be the vector space of realvalued functi...
 8.4.20: Let V be a fourdimensional vector space with basis B, let W be a s...
 8.4.21: In each part, fill in the missing part of the equation. (a) [T2 T1]...
 8.4.22: Prove that if T : V W is the zero transformation, then the matrix f...
 8.4.23: Prove that if B and B are the standard bases for Rn and Rm, respect...
 8.4.TF: TF. In parts (a)(e) determine whether the statement is true or fals...
Solutions for Chapter 8.4: Matrices for General LinearTransformations
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 8.4: Matrices for General LinearTransformations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.4: Matrices for General LinearTransformations includes 24 full stepbystep solutions. 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. Since 24 problems in chapter 8.4: Matrices for General LinearTransformations have been answered, more than 16685 students have viewed full stepbystep solutions from this chapter.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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 matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.