 2.8.1: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.2: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.3: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.4: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.5: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.6: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.7: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.8: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.9: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.10: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.11: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.12: !11 Exerci;es 1 12. find a genemting set for the range of each lin...
 2.8.13: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.14: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.15: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.16: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.17: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.18: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.19: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.20: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.21: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.22: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.23: In Exercises 13 23, find tt genertlling set for 1he null space of ...
 2.8.24: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.25: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.26: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.27: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.28: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.29: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.30: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.31: In rercises 24 3 /,find The swndard mmrix of each linear 1/'tmsfom...
 2.8.32: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.33: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.34: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.35: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.36: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.37: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.38: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.39: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.40: In Exercises 3240,find the sTandard matrix of each linear Transfor...
 2.8.41: A linear transformation with codomain 'R"' is onto if and only if t...
 2.8.42: A linear transformation is onto if and o nly if the columns of its ...
 2.8.43: A linear transform
 2.8.44: A linear transfo rmation is onetoone if and only if every vector ...
 2.8.45: A linear transformation is onetoone if and only if its null space...
 2.8.46: A linear transformation is invertible if and only if its standard m...
 2.8.47: The system Ax = b is consistent for all b if and only if the transf...
 2.8.48: Let A be an m x 11 matrix. The system Ax = b is consistent for all ...
 2.8.49: A function is onto if its mnge equals its domain.
 2.8.50: A function is onto if its mnge equals its codomain.
 2.8.51: The set {T(et). T(e2) . .. . , T(e,.) } is a generating set for the...
 2.8.52: A linear transformation T: n"  n"' is onto if and only if the rank...
 2.8.53: The null space of a linear transformation T: n > nm is the set of ...
 2.8.54: A function T: n > nm is onetoone if the only vector v in n" whos...
 2.8.55: A linear transformation T: n" > n"' is onetoone if and only if t...
 2.8.56: If the composition UT of two linear transformations T: n"+ nm and ...
 2.8.57: The composition of linear transformations is a linear transformation.
 2.8.58: If T: n"  nm and U : 'RJ'  n" are linear transformations with sta...
 2.8.59: For every invertible linear transformation T. the function r l is ...
 2.8.60: If A is the standard matrix of an invertible linear transformation ...
 2.8.61: Suppose that T: n2 + n2 is the linear transformation that rotates ...
 2.8.62: Suppose that T: n2 + n2 is the reflection of n2 about the xaxis. ...
 2.8.63: Define T : n2 + n2 by T ([::~]) = [~l which is the projection of [...
 2.8.64: Define T: n3 ..... n3 by T ([~:]) the projection of [; ] o n the z ...
 2.8.65: Define T: n3 . n3 by T [;~ ]) [;~ J which is the projection of [;:...
 2.8.66: efine T: n3 ~ n 3 by T ([;~ ]) [ _::] (See Exercise 92 in Section 2...
 2.8.67: Suppose that T: n2 > n2 is linear and has the property that T(et) = m
 2.8.68: Suppose that T: n 2 > n2 is linear and has the property that T(e 1...
 2.8.69: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.70: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.71: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.72: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.73: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.74: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.75: Exercises 6975 are co11cemed with the li11ear tra11sj'ormations T:...
 2.8.76: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.77: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.78: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.79: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.80: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.81: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.82: Exerdses 76 82 are concemed ll'itlr tire linear transformations T :...
 2.8.83: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.84: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.85: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.86: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.87: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.88: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.89: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.90: In Exurius 8390. an illl't'rtibl~ lin~ar trtmsfonnation T is d~fin...
 2.8.91: Prove lhm the composition of two onetoone linear trans formation ...
 2.8.92: Prove th;rl if 1wo linear transformations are onto. then I heir com...
 2.8.93: In n2 sho" that the composition of two reflectton:. about the A ax...
 2.8.94: In n2 show that a rellcction about the yaxi\ follo\\ed by :t rotat...
 2.8.95: In n2 show that the composition of the proJection on the .\axis fo...
 2.8.96: Prove that the composition of two shear transforrmuions is a shear ...
 2.8.97: Suppose that T : n + n'" is linear and oneloone. Let (v,, ' '! ....
 2.8.98: Use Theorem 2.12 to prove that matrix multiplication is associative.
 2.8.99: The linear tran;,formalions T. U: R 4 + n4 are defined
 2.8.100: Define the linear transformation T: R.4 > R.4 by the rule (a) Find...
Solutions for Chapter 2.8: MATRICES AND LINEAR TRANSFORMATIONS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 2.8: MATRICES AND LINEAR TRANSFORMATIONS
Get Full SolutionsElementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. Since 100 problems in chapter 2.8: MATRICES AND LINEAR TRANSFORMATIONS have been answered, more than 25395 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Chapter 2.8: MATRICES AND LINEAR TRANSFORMATIONS includes 100 full stepbystep solutions.

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

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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 •

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

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

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.