 6.2.1: Let L : R2 ....... R" be the linea! transformation defined by (a ) ...
 6.2.2: Let L : R2 ;. R2 be the linea! operator defined by I' J Is [2 3  ...
 6.2.3: Let L : ~ ....... R2 be the linear tmnsfonnation defined by Ie) Fin...
 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: LeI L:"2 > H.l be the linear trnllsformulion defined by L(a/ 2 +bl ...
 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 6.6 for the following linear transfonnalions: (a) L:...
 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...
Solutions for Chapter 6.2: Kernel and Range of a linear Transformation
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
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. Since 30 problems in chapter 6.2: Kernel and Range of a linear Transformation have been answered, more than 4267 students have viewed full stepbystep solutions from this chapter. Chapter 6.2: Kernel and Range of a linear Transformation includes 30 full stepbystep solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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.

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.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

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