 2.1: A symmetric matrix equals its transpose
 2.2: If a symmetric matrix is written in block fom1, then the blocks are...
 2.3: The product of square matrices is always defined.
 2.4: The transpose of an invertible matrix is invertible.
 2.5: It is possible for an invertible matrix to have two distinct inverses.
 2.6: The sum of an invertible matrix and its inverse is the zero matrix.
 2.7: The columns of an invertible matrix are linearly independent.
 2.8: If a matrix is invertible, then its rank equals the number of its r...
 2.9: A matrix is invertible if and only if tt> reduced ro" echelon form ...
 2.10: If A i> an n x 11 matrix and the 'Y'tem Ax = b is consistent for so...
 2.11: The range of a linear tmnsformation i> contained in the codomai n o...
 2.12: 11e null s pace of a linear transformation is contained in the codo...
 2.13: Linear tr:msformations preserve linear combinations.
 2.14: Linear transformations preserve linearly independent sets.
 2.15: Every linear transformation has a >tandard matrix.
 2.16: The zero transformation is the only linear transformation whose sta...
 2.17: If a linear transformation is onetoone. then it is inverttble.
 2.18: If a linear transformation is onto. then it' range equals its codom...
 2.19: If a linear transformation is onetoone. then its range consists e...
 2.20: If a linear transformation is onto. then the rows of its standard m...
 2.21: If a linear transformation is onetoone. then the columns of its s...
 2.22: Determine whether each phrase i; a misuse of terminology. If so. ex...
 2.23: Let A be an m x 11 matrix and /J be a p x q matrix. (a) Under what ...
 2.24: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.25: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.26: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.27: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.28: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.29: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.30: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.31: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.32: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.33: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.34: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.35: In Exercius 24 35, uu the giten matrias to com1mte t'aclt eX/J/l'S...
 2.36: In E.urrisu 36 and 37. compflle tlte pro 6[j 0 3 i][ I I 0 !] I 2 ...
 2.37: In E.urrisu 36 and 37. compflle tlte pro [ /z 1htll
 2.38: In E.utt"itef 38 and 39, dnermine whether elldtmlltrix i.r iiii'UIi...
 2.39: In E.utt"itef 38 and 39, dnermine whether elldtmlltrix i.r iiii'UIi...
 2.40: Let A and 8 be square matrices of the same siLC. Prove that if the ...
 2.41: Let A nnd 8 be square matrices of the same size. Prove that if the ...
 2.42: Give examples of 2 x 2 matrices A and 8 such that A and 8 are inven...
 2.43: /11 .un1tt't 43 tmd 44. systems ofet/lllltimu art' gnm. Fir 2.1, + ...
 2.44: /11 .un1tt't 43 tmd 44. systems ofet/lllltimu art' gnm. Fir x, + xz...
 2.45: Suppose that the reduced row echelon form R and three columns of A ...
 2.46: E.rercist's 4649 refer to Ilte following matrices: A = [! ~ ~] an...
 2.47: E.rercist's 4649 refer to Ilte following matrices: A = [! ~ ~] an...
 2.48: E.rercist's 4649 refer to Ilte following matrices: A = [! ~ ~] an...
 2.49: E.rercist's 4649 refer to Ilte following matrices: A = [! ~ ~] an...
 2.50: In Exercises 50 53. a linear transformmion is given. Compllle its ...
 2.51: In Exercises 50 53. a linear transformmion is given. Compllle its ...
 2.52: In Exercises 50 53. a linear transformmion is given. Compllle its ...
 2.53: In Exercises 50 53. a linear transformmion is given. Compllle its ...
 2.54: In Exercises 54 57. a function T: n" + n"' is given. Either prov...
 2.55: In Exercises 54 57. a function T: n" + n"' is given. Either prov...
 2.56: In Exercises 54 57. a function T: n" + n"' is given. Either prov...
 2.57: In Exercises 54 57. a function T: n" + n"' is given. Either prov...
 2.58: In Exercises 58 and 59. find a generating set for the range of each...
 2.59: In Exercises 58 and 59. find a generating set for the range of each...
 2.60: In Exercises 60 and 61. find a generating set for the null space of...
 2.61: In Exercises 60 and 61. find a generating set for the null space of...
 2.62: In Exercises 62 and 63. find rite standard marrLr of each linecrr r...
 2.63: In Exercises 62 and 63. find rite standard marrLr of each linecrr r...
 2.64: In Exercises 64 and 65, find the standard matrix of each linear tra...
 2.65: In Exercises 64 and 65, find the standard matrix of each linear tra...
 2.66: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.67: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.68: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.69: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.70: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.71: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.72: Erercises 66 72 are concerned wirh rhe linear Transformations T: n...
 2.73: In Exercises 73 and 74, an invertible linear Transformation T is de...
 2.74: In Exercises 73 and 74, an invertible linear Transformation T is de...
Solutions for Chapter 2: MATRICES AND LINEAR TRANSFORMATIONS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 2: MATRICES AND LINEAR TRANSFORMATIONS
Get Full SolutionsSince 74 problems in chapter 2: MATRICES AND LINEAR TRANSFORMATIONS have been answered, more than 25498 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2: MATRICES AND LINEAR TRANSFORMATIONS includes 74 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

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).

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.