 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 6697 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.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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 •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.