 6.2.1: Let L : R2 ....... R" be the linea! transformation defined byL ([::...
 6.2.2: Let L : R2 ;. R2 be the linea! operator defined byL ([::;]) ~ [; 4...
 6.2.3: Let L : ~ ....... R2 be the linear tmnsfonnation defined byL ([ II ...
 6.2.4: Let L : R2 + R3 be the linear transformation defined by L([1I 1 11...
 6.2.5: Let L : ~ + R3 be the linear trnnsfonnation defined by L([1I1 112 ...
 6.2.6: Let L: P2 + P3 be the linear trnnsfonmltion defined by L(p(l = 12,...
 6.2.7: Let L : M2j __ Mn be the lirrar transformation defined by 2 I] A I...
 6.2.8: Let L: 1'2 + PI be the linear transform.'1tion defined by L(tlt 2 ...
 6.2.9: LeI L:"2 > H.l be the linear trnllsformulion defined by L(a/ 2 +bl ...
 6.2.10: Let L : Atn ...... Mn be the lirrar transformation defined by L(A)=...
 6.2.11: lei L: M 22 + M n be the linear operator defined by ([a b]) [a +b ...
 6.2.12: leI L : V __ IV be a linear transfoml:ltion. (a) Show that dimrange...
 6.2.13: Verify Theorem 6.6 for the following linear transfonnalions: (a) L:...
 6.2.14: Verify Theorem 6.5 for the linear InUlsfonnation gi\"en in Exercise...
 6.2.15: Lei A be an 11/ X " matrix. and consider the linear transfomlalion ...
 6.2.16: LeI L : RS + R' be the linear transformation defin~d by L([]){ 0 ...
 6.2.17: Let L : Rj ...... R3 be the linear tr.lIIsfonnation defined by L(e ...
 6.2.18: Let L : V ,I IV be a linear tmnsfonnation. and let dim V = dim I...
 6.2.19: Let L: 1(3 ,I 1(3 be defined by (a) Prove that L is invertible
 6.2.20: Let L : V __ IV be a linear transfonnation. and let 5 = {VI. V! ......
 6.2.21: Find the dimension of the solution space for the followmg homogeneo...
 6.2.22: Find a linear transfonnation L : 1(2 ,I R 3 such that S = ([I  ...
 6.2.23: Let L : f( l __ 1 be the Imeat transformation defined by
 6.2.24: Let L V + IV be a linear transfonnation. Prove that L is onetoon...
 6.2.25: Let L: R'; ___ 1(6 be a linear transfonnation. (II) If dim kef L = ...
 6.2.26: LeI L : V )0 RS be a linear transfonnation. (a) [f L is onto and ...
 6.2.27: Let L be the linear transformation defined in Exercise 24. Section ...
 6.2.28: Let L be the linear transformation defined in Exercise 25 . Section...
 6.2.29: Prove Corollary 6.1.
 6.2.30: Let L : RIO ..... Rm be a linear transformation defined by L(x) = A...
 6.2.31: Prove Corollary 6.2.
Solutions for Chapter 6.2: Kernel and Range of a Linear Transformation
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 6.2: Kernel and Range of a Linear Transformation
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. 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: 9780132296540. Chapter 6.2: Kernel and Range of a Linear Transformation includes 31 full stepbystep solutions. Since 31 problems in chapter 6.2: Kernel and Range of a Linear Transformation have been answered, more than 12008 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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.

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

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.

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

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

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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.