Table of Symbols

Symbol Meaning Reference ∅ Empty set Page 10 ∈ Member symbol Page 10 ⊆ Subset symbol Page 10 ⊂ Proper subset symbol Page 10 ∩ Intersection symbol Page 10 ∪ Union symbol Page 10 ⊗ Tensor symbol Page 136 ≈ Approximate equality sign Page 79 −−→ PQ Displacement vector Page 147 | z | Absolute value of complex z Page 13 | A | determinant of matrix A Page 115 || u || Norm of vector u Page 212 || u ||p p-norm of vector u Page 306 u · v Standard inner product Page 216 u, v Inner product Page 312 Acof Cofactor matrix of A Page 122 adj A Adjoint of matrix A Page 122 A∗ Conjugate (Hermitian) transpose of matrix A Page 91 AT Transpose of matrix A Page 91 C(A) Column space of matrix A Page 183 cond(A) Condition number of matrix A Page 344 C[a, b] Function space Page 151 C Complex numbers a + bi Page 12 Cn Standard complex vector space Page 149 compv u Component Page 224 z Complex conjugate of z Page 13 δij Kronecker delta Page 65 dim V Dimension of space V Page 194 det A Determinant of A Page 115 domain(T ) Domain of operator T Page 189 diag{λ1,λ2,...,λn} Diagonal matrix with λ1,λ2,...,λn along diagonal Page 103 Eij Elementary operation switch ith and jth rows Page 25 356 Table of Symbols

Symbol Meaning Reference Ei(c) Elementary operation multiply ith row by c Page 25 Eij (d) Elementary operation add d times jth row to ith row Page 25 Eλ(A) Eigenspace Page 254 Hv Householder matrix Page 237 I,In Identity matrix, n × n identity Page 65 idV Identity function for V Page 156 (z) Imaginary part of z Page 12 ker(T ) Kernel of operator T Page 188 Mij (A) Minor of A Page 117 M(A) Matrix of minors of A Page 122 max{a1,a2,...,am} Maximum value Page 40 min{a1,a2,...,am} Minimum value Page 40 N (A) Null space of matrix A Page 184 N Natural numbers 1, 2,... Page 10 null A Nullity of matrix A Page 39 P Space of polynomials of any degree Page 163 Pn Space of polynomials of degree ≤ n Page 163 projv u Projection vector along a vector Page 224 projV u Projection vector into subspace Page 327 Q Rational numbers a/b Page 11 (z) Real part of z Page 12 R(A) Row space of matrix A Page 184 R(θ) Rotation matrix Page 178 R Real numbers Page 11 Rn Standard real vector space Page 147 Rm,n Space of m × n real matrices Page 151 TA Matrix operator associated with A Page 72 range(T ) Range of operator T Page 189 rank A Rank of matrix A Page 39 ρ(A) Spectral radius of A Page 273 span{S} Span of vectors in S Page 164 sup{E} Supremum of set E of reals Page 343 target(T ) Target of operator T Page 189 [T ]B,C Matrix of operator T Page 243 V ⊥ Orthogonal complement of V Page 333 Z Integers 0, ±1, ±2,... Page 11 Solutions to Selected Exercises

Section 1.1, Page 8

1 (a) x = −1, y =1(b)x =2,y = −2, a11 =1,a12 = −3, b1 =1,a21 =0, z =1(c)x =2,y =1 a22 =1,b2 =5 3 (a) linear, x − y − z = −2, 3x − y =4 47 − − 47 − 7 25 y1 y2 =0, y1 + 25 y2 y3 =0, (b) nonlinear (c) linear, x +4y =0, − 47 − − 47 y2 + 25 y3 y4 =0, y3 + 25 y4 =0 2x − y =0,x + y =2 9 p1 =0.2p1 +0.1p2 +0.4p3, p2 =0.3p1 + 5 (a) m =3,n =3,a11 =1,a12 = −2, 0.3p2 +0.2p3, p3 =0.1p1 +0.2p2 +0.1p3 a13 =1,b1 =2,a21 =0,a22 =1, a23 =0,b2 =1,a31 = −1, a32 =0, 13 Counting inﬂow as positive, the equa- a33 =1,b3 =1(b)m =2,n =2, tion for vertex v1 is x1 − x4 − x5 =0.

Section 1.2, Page 19 √ √ 1 (a) {0, 1} (b) {x | x ∈ Z and x>1} 11 (a) z = −1 ± 11 i,(b)z = ± 3 + 1 i 2 √ √2 √ √ 2 2 (c) {x | x ∈ Z and x ≤−1} (d) − − (c) z =1± ( 2 2+2 − 2 2 2 i) (d) {0, 1, 2,...} (e) A 2 2 ±2i √ 3πi/2 πi/4 2πi/3 3 (a) e √(b) 2e (c) 2e (d) 13 (a) Circle of radius 2, center at ori- e0ior 1 (e) 2 2e7πi/4 (f) 2eπi/2 (g) eπe0i gin (b) (z) = 0, the imaginary axis or eπ (c) Interior of circle of radius 1, center at z =2. 5 (a)1+8i(b)10+10i (c) 3 + 4 i(d) 5 5 − − − − 3 − 4 i (e) 42 + 7i 15 2 + 4i+1 3i = 2 4i+1+3i = 3 i 5 5 and (2+4i)+(1− 3i) = 3+i=3− i √ √ 6 − 8 ± ± ± − − − 7 (a) 5 5 i, (b) 2 i 2, (c) z =1 17 z =1 i, (z (1 + i)) (z (1 i)) = (d) z = −1, ±i z2 − 2z +2

√ √ 2 1 1 1 πi/4 − − 21 Use |z| = zz¯ and z1z2 = z1z2. 9 (a) 2 + 2 i= 2 √2e (b) 1 i 3= 4πi/3 − 2πi/3 − 1 n 2e (c) 1+i 3=2e (d) 2 i= 24 Write p (w)=a0 +a1w+···+anw = 1 3πi/2 π/4 π/4 πi/2 2 e (e) ie =e e 0 and conjugate both sides. 358 Solutions to Selected Exercises

Section 1.3, Page 30

× 2 1 −1 1 1 (a) Size 2 4, a11 = a14 = a23 = a24 = 9 (a) x1 = 3 b1 + 3 b2, x2 = 3 b1 + 3 b2 1, a12 = −1, a21 = −2, a13 = a22 =2 (b) Inconsistent if b2 =2 b1, otherwise (b) Size 3 × 2, a11 =0,a12 =1,a21 =2, solution is x1 = b1 + x2 and x2 arbi- − 1 − a22 = 1, a31 =0,a32 =2(c)Size trary. (c) x1 = 4 (2b1 + b2)(1 i), x2 = × − × 1 − − 2 1,a11 = 2, a21 =3(d)Size1 1, 4 (ib2 2b1)(1 i) a11 =1+i 11 The only solution is the trivial solu- 23 7 tion p1 =0,p2 =0,andp3 =0,which 3 (a) 2×3 augmented matrix , 12−2 has nonnegative entries. x = 20, y = −11 (b) 3×4 augmented ma- 13 Augmented matrix with three right- 36−1 −4 102−1 −3 hand sides reduces to trix −2 −413, x1 = −1−2x2, x2 011−1 −3

0011 given solutions (a) x1 =2,x2 =1(b) free, x =1, (c) 3 × 3 augmented matrix 3 x1 = −1, x2 = −1(c)x1 = −3, x2 = −3. 11−2 52 5 , x1 =3,x2 = −5 15 (a) x =0,y = 0 or divide by xy and 12−7 get y = −8/5, x =4/7(b)Eithertwo of x, y, z are zero and the other arbitrary

5 (a) x1 =1− x2, x3 = −1, x2 free or all are nonzero, divide by xyz and ob- − (b) x1 = −1 − 2x2, x3 = −2, x4 =3, tain x = 2z, y = z,andz is arbitrary x2 free (c) x1 =3− 2x3, x2 = −1 − x3, nonzero. x free (d) x =1+2 i, x =1− 1 i 3 1 3 2 3 17 Suppose not and consider such a so- (e) x = 7 x , x = −2 x , x = 6 x , 1 11 4 2 11 4 3 11 4 lution (x, y, z, w). At least one variable x4 free is positive and largest. Now examine the equation corresponding to that variable. 7 (a) x1 =4,x3 = −2, x2 free (b) x1 = 1, x2 =2,x3 = 2 (c) Inconsistent system 19 (a) Equation for x2 =1/2isa + b · 2 1/2 (d) x1 =1,x2 and x3 free 1/2+c · (1/2) = e . Section 1.4, Page 42 1 (b) and (d) are in reduced row form, 1 1001 nullity 1 (c) E12, E1 , , (a), (e), (f), and (h) are in reduced row 2 0101 echelon form. Leading entries (a) (1, 1), rank 2, nullity 2 (d) E 1 , E (−4), 1 2 21 (3, 3) (b) (1, 1), (2, 2), (3, 4) (c) (1, 2), 10 3 (2, 1) (d) (1, 1), (2, 2) (e) (1, 1) (f) (1, 1), E31 (−2), E32 (1), E12 (−2), 01−1 , (2, 2), (3, 3) (g) (1, 2) (h) (1, 1) 00 0 rank 2, nullity 1 (e) E , E (−2), 3 (a) 3 (b) 0 (c) 3, (d) 1 (e) 1 12 21 110 22 E 1 , E (2) 9 ,rank2,nul- − − − 2 9 12 001 2 5 (a) E21 ( 1), E31 ( 2), E32 ( 1), 9 5 − − 10 2 lity 2 (f) E12, E31 ( 1), E23, E2 ( 1), E 1 , E (1), 01 1 ,rank2,nul- E (3), E −1 , E (1), E (−1), 2 4 12 2 32 3 4 23 13 000 100 − − − lity 1 (b) E21 (1),E23 ( 15), E13 ( 9), E12 ( 2), 010 , rank 3, nullity 0 17 100 3 001 − 1 − E12 ( 1), E1 3 , 010 33 ,rank3, 001 2 Solutions to Selected Exercises 359

7 Systems are not equivalent since of right-hand side, so system is always (b) has trivial solution, (a) does not. consistent. Solution is x1 = −a+2b−c+ − 1 − 1 − (a) rank A =2,rank(A)=4x4, x2 = b+a+ 2 c 2x4, x3 = 2 c x4, x4 free. 2, {(−1+x3 + x4 , 3 − 2x2,x3,x4) | x3, x4 ∈ R} (b) rank A =3,rank(A)= 15 (a) 3 (b) solution set (c) E23 (−5) 3, {(−2x2,x2, 0, 0) | x2 ∈ R} (d) 0 or 1 9 0 < rank (A) < 3 17 (a) false, (b) true, (c) false, (d) false, 11 (a) Inﬁnitely many solutions for all c (e) false (b) Inconsistent for all c (c) Inconsistent if c = −2, unique solution otherwise 20 Consider what you need to do to go 13 Rank of augmented matrix equals from reduced row form to reduced row rank of coeﬃcient matrix independently echelon form.

Section 1.5, Page 54 ⎡ ⎤ 130420 0 4 Work of jth stage: j + ⎢ 001200 0⎥ 2[(n − 1) + (n − 2) + ···+(n − j)]. 3 (a) rank A =3,(b)⎣ 1 ⎦ . 000001 3 000000 0 Section 2.1, Page 60 −21−1 4 1 2 0 1 (a) (b) (c) 5 (a) x + y + z −11 1 −1 2 0 −1 74−1 3 2 28 1 −1 (d) not possible (e) 10 4 4 (b) x +y (c) x 0 +y 0 + 63 2 3 24 0 1 1 x − 2+4y 0 1 −3 0 (f) 3x − 2+y z −1 (d) x 4 + y 0 + z 1 −1 5 0 2 −1 7 a = −2 , b = 2 , c = −4 3 3 3 ab 10 01 9 = a + b + cd 00 00 00 00 c + d 10 01 −12−3 −1 −3 −2 11 A +(B + C)= = 3 (a) not possible (b) − − 51 5 4 14 02−2 0 −1 −1 (A + B)+C, A+B = = B+A (c) − (d) not possible 42 5 10 2 58 3 16 Solve for A in terms of B with the (e) − 1 13 5 6 ﬁrst equation and deduce B = 4 I. 360 Solutions to Selected Exercises

Section 2.2, Page 68 68 1 (a) [11 + 3i], (b) , (c) impossi- 13 (b) is not nilpotent, the others are. 34 01 00 15 + 3i 20 + 4i 15 A = and B = are ble (d) impossible (e) 00 10 −3 −4 01 (f) impossible (g) [10] (h) impossible nilpotent, A + B = is not nilpo- ⎡ ⎤ 10 x1 tent. 1 −240 3 ⎢ x2 ⎥ −−−−−−→ 3 (a) 01−10 ⎣ ⎦ = 2 −111 1 −1 −1 x E1(−1) −1 004 3 1 17 uv = 000 00 0 , x E31(−2) 4 −222 00 0 − − 1 1 3 x 3 so rank uv =1 (b) 224 y = 10 − 48 101 z 3 19 A (BC)= =(AB) C, 1 −30 x −1 12 (c) 020 y = 0 016 c (AB)= =(cA) B = A (cB) −130 z 0 04 10 −11 x 3 ab 23 Let A = and try simple B like 5 2 −4 −2 y = 1 cd − 42 2 z 2 10 . 101 2 101 1 00 7 (a) 113 −4 (b) 311 −1 27 Let Am×n =[aij ]and⎡ Bm×n =[⎤bij ]. 011 −3 110 2i a11 0 ···0 10 −11 x1 ⎢ ⎥ If b =[1, 0,...,0]T , ⎣ . . ⎦ = (c) −42−2 −3x2 . . 42−2 x3 am1 0 ···0 ⎡ ⎤ b11 0 ···0 34 1 −2 ⎢ ⎥ 9 f (A)= , g (A)= − , ⎣ . . ⎦ 25 10 . . so a11 = b11,etc.Bysim- −1 −6 bm1 0 ···0 h (A)= −3 −4 ilar computations, you can show that for each i, j, aij = bij . 38 11 A2 = , BA = 616 , AC = 411 11 −172 , AD = − , BC = [22], 16 293 24 CB = , BD = −2144 10 20 Section 2.3, Page 84 1 (±1, ±1) map to (a) (±1, ∓1) 11 (b) ± 7 , 1 , ± −1 , 7 (c) ± (1, 1), 3 (a) A = 20 (b) nonlinear 5 5 5 5 4 −1 ± (1, −1) (d) ± (2, −1), ± (0, 1) −110 002 (c) (d) 001 −100 011 Solutions to Selected Exercises 361 √ ⎡ ⎤ 3 √−2 00001 5 Operator is TA, A = ,and ⎢ 10001⎥ √12 3 ⎢ ⎥ matrix is ⎢ 01000⎥ and picture: 3 √−1 ⎣ ⎦ in reverse order TB , B = . 01100 22 3 00010

v1 e1 v2

e2 e3 7 (d) is the only candidate and the only v5 ﬁxed point is (0, 0, 0). e7 e5 e4

e6 v4 v3 9 (a), (b) and (c) are Markov. First 3 5 and second states are (a) (0.2, 0.2, 0.6), ak+3 − 2 − ak+2 1 1 2 2 (0.08, 0.68, 0.24) (b) 2 (0, 1, 1), 2 (1, 1, 0) 13 (a) ak+2 = 10 0 ak+1 (c) (0.4, 0.3, 0.4), (0.26, 0.08, 0.66) ak+1 01 0 ak (d) (0, 0.25, 0.25), (0.225, 0, 0.15) a 1 −2 a 1 (b) k+2 = k+1 + ak+1 10 ak 0 15 Points on a nonvertical line through the origin have the form (x, mx). 11 Powers of vertices 1–5 are 2, 4, 3, 5, 3, respectively. Graph 17 Use Exercise 27 of Section 2 and the is dominance directed, adjacency deﬁnition of matrix operator.

Section 2.4, Page 97 100 001 100 110 − 11 0 E 1 E − (c) = 2 2 21 ( 1) − 1 (a) 013 (b) 010 (c) 010 001 11 2 001 100 002 10 −2 (d) = E 1 × 100 01 1 (1 + i) 1+i (d) 010(e) E (3) (f) E (−a) 2 12 31 01+i i 0 −11 E 12 10−2 (g) E2 (3) (h) E31 (2) 7 (a) strictly upper triangular, tridiago- 3 (a) add 3 times third row to second nal (b) upper triangular (c) upper and (b) switch first and third rows (c) mul- lower triangular, scalar (d) upper and tiply third row by 2 (d) add −1times lower triangular, diagonal (e) lower tri- second row to third (e) add 3 times sec- 20 angular, tridiagonal ond row to first (f) add −a times first 31 row to third (g) multiply second row by 3 (h) add 2 times first row to third 02I3 9 A = with C =[4, 1], CD 12 0 −I2 5 (a) I2 = E12(−2)E21(−1) (b) D =[2, 1, 3], B = with 13 EF 10−1 110 00 12 01 1 = E12 (−1) E32 (−2) 011 E = 22 and F = −11 , 00 0 022 11 32 362 Solutions to Selected Exercises 0+2I3E 0(−I2)+2I3F 15 (a) true (b) false (c) false (d) true AB = − = C 0+DE C⎡ ( I2)+DF⎤ (e) false 00 2 4 17 Q(x, y, z)=xT Ax with x =[x, y, z]T 2E 2F ⎢ 44−22⎥ = ⎣ ⎦ 22−6 DE −C + DF 22 6 4 and A = 01 4 55 6 10 00 1 T 11 102 121 ∗ 4 −2+4i 19 A A = − − = 13 (a) (1, −3, 2), (1, −3, 2), not sym- 2 4i 14 20 1 ∗ ∗ ∗ 93− 6i metric or Hermitian (b) , (A A) and AA = = 13−4 3 + 6i 9 ∗ ∗ 20 1 (AA ) − , not symmetric or Hermitian 13 4 22 Since A and C are square, you can 1 −i 1i confirm that block multiplication applies (c) , − , Hermitian, not i2 i2 and use it to square M. 113 113 29 Compare (i, j)th entries of each side. symmetric (d) 100 , 100 ,sym- 302 302 32 Substitute the expressions for A into metric and Hermitian the right-hand sides and simplify them.

Section 2.5, Page 111 1 1 1 − − 21−2 2 2 2 1 i 1 (a) 0 1 0 (b) 4 (c) does 9 Both sides give 1 2 −12. 2 0 1 4 1 1 1 4 −212 2 2 2 ⎡ ⎤ 1 − 1 − 1 1 2 2 2 2 ⎢ 011−1 ⎥ notexist(DNE)(d)⎣ ⎦ 18 12 9 001 0 2 11 Both sides give 1 02−1 . 0001 12 −60 3 cos θ sin θ (e) − sin θ cos θ −1 −k 13 (a) any k, 23 2 −3 20 01 3 (a) , , − − − 12 12 11 101 − − − 1 − − 36 1 1 67 1 (b) k =1, k−1 kk 11 − 1 − k 0 −1 (b) 21 1 , 15 231 , 0 ⎡ ⎤ −1 00 1 0015 1 100k 11 −21 3 ⎢ 0 −100⎥ 1 (c) k =0, ⎣ − ⎦ (c) , 3 , 1 52 5 −1 −5 006 0 1 000k 5 (a) E21 (−3) (b) E2(−1/2) (c) E21 (−1) E13 (d) E12 (1) E23 (1) −1 10 −10 (e) E3 E1 (−1) E21 (i) 15 Let A = , B = ,so 3 01 0 −1 − − 00 1 3 31 both invertible, but A + B = ,not 7 00−6 −4 00 0 −1 −5 −3 invertible. Solutions to Selected Exercises 363 01−2 x2 +sin(πxy) − 1 − 2 2 x+y − , JF (x)= 17 (a) N = 00 1 , I + N + N + x + y + e 3 00 0 2x + cos (πxy) ,πy cos (πxy) πx − 11 3 000 1+ex+y, 2y + ex+y N 3 = 01−1 (b) N = 000 , − 00 1 100 21 Move constant term to right-hand 100 side and factor A on left. I + N = 010 −101 24 Multiplication by elementary matrices does not change rank. 1.00001 19 x =(x, y), x(9) ≈ , − −1 0.99999 29 Assume M has the same form as (9) −6 −1.3422 M and solve for the blocks in M using F x ≈ 10 , F (x)= − 2.0226 MM 1 = I. Section 2.6, Page 125 − − − 10 0 1 (a) A11 = 1, A12 = 2, A21 = 2, 4 −1 − 11 (a) (b) − 1 1 − 1 A22 =1(b)A11 =1,A12 =0,A21 = 3, −31 2 2 2 −10 1 A22 =1(c)A22 = 4, all others are 0 − −1 −4 −2 (d)A11 =1,A12 =0,A21 = 1+i, −1i (c) 0 −1 −1 (d) A22 =1 −2i −1 110

1 3 All except (c) are invertible. (a) 3, (b) 13 (a) x =5,y =1(b)x1 = 4 (b1 + b2), 1 − −7 5 1+i, (c) 0, (d) −70, (e) 2i x2 = 2 (b1 b2)(c)x1 = 6 , x2 = 3 , 11 x3 = 2 17 Use elementary operations to clear 5 Determinants of A and AT are (a) −5 the ﬁrst column and factor out as many (b) 5 (c) 1 (d) 1 (xj − xk) as possible in the resulting determinant. 7 (a) a = 0 and b =1(b) c =1(c)any θ 19 Take determinants of both sides of the identity AA−1 = I. −2 −22 21 Factor a term of −2 out of each row. 9 (a) 44−4 ,03,3 (b) What remains? −3 −33 23 Use row operations to make the diag- −10−3 2 −3 onal submatrices triangular. 0 −40, −4I3 (c) ,5I2 11 2 −101 24 Show that Jn = In, which narrows (d) 04,4,04,4 down det Jn.

Section 2.7, Page 143 100 2 −11 1 L = 110 , U = 04−3 , 211 00−1 − 1 − − (a) x =(1, 2, 2) (b) x = 4 (3, 6, 4) 1 − − 1 (c) x = 4 (3, 2, 4) (d) x = 8 (3, 6, 8) 364 Solutions to Selected Exercises ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2 −10 00 0 200−100 3 0 0100 x11 2 ⎢ ⎥ ⎢ − − − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 100000⎥ ⎢ 442 2 2 1 ⎥ ⎢ 2 4 1010⎥ ⎢ x21 ⎥ ⎢ 1 ⎥ ⎢ 4 −24−22−1 ⎥ ⎢ 202−10−1 ⎥ ⎢ 1 0 3001⎥ ⎢ x ⎥ ⎢ 1 ⎥ 3 ⎢ ⎥,⎢ ⎥ 5 ⎢ ⎥ ⎢ 31 ⎥ = ⎢ ⎥ ⎢ 202010⎥ ⎢ 100 0 0 0⎥ ⎢ −1 0 0100⎥ ⎢ x ⎥ ⎢ −1 ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ 12 ⎦ ⎣ ⎦ 2 −10 02−1 221 0 0 0 0 −1 0221 x22 0 − ⎡ 100010 ⎤101 0 0 0 001101 x32 3 020−100 7 If so, each factor must be nonsingular. ⎢ −240000⎥ ⎢ ⎥ Check the (1, 1)th entry of an LU prod- ⎢ 0 −1010−1 ⎥ ⎢ ⎥for (a), (b), (c), uct. ⎢ 1 −2 −121−2 ⎥ ⎣ ⎦ 0 −20002 13 For matrices M,N, block arith- 2 −400−24 metic gives MN =[Mn1,...,Mnn]. (d) same as (c) Use this to show that vec (MN)= (I ⊗ M)vec(N). Also, Mnj = n1j m1 + ··· + npj mp. Use this to show that vec (MN)= N T ⊗ I vec (M). Then apply these to AXB = A (XB).

Section 3.1, Page 158

1 (a)(−2, 3, 1) (b) (6, 4, 9) 13 (a) linear, range not V (b) not linear, (c) linear, range is V (d) linear, range not 3 V is a vector space. V 5 V is not a vector space because it is 15 (a) identity operator is linear and −1 not closed under scalar multiplication. invertible, (idV ) =idV

7 V is not a vector space because it is 17 Mx =(x1 +2,x2 − 1,x3 +3, 1), so not closed under vector addition or scalar action of M is to translate the point − multiplication. in direction of vector (2, −1, 3). M 1 = I3 −t 9 V is a vector space. (think inverse action) 0 1 10 0 19 Write c0 = c (0 + 0)=c0 + c0 11 (a) T = TA, A = − , linear 12 4 by identity and distributive laws. Add with range C (A)=R2, equal to target − (c0) to both sides. 010 (b) not linear (c) T = TA, A = , 27 Use the fact that T ◦T = T and 010 A B AB T =id linear with range C (A) = span {(1, 1)}, I not equal to target (d) not linear (e) not 28 Use the fact that TA ◦TB = TAB and linear do matrix arithmetic.

Section 3.2, Page 168

1 W is not a subspace of V because W 5 W is a subspace. is not closed under addition and scalar multiplication. 7 Not a subspace, since W doesn’t con- tain the zero element.

3 W is a subspace. 9 W is a subspace of V. Solutions to Selected Exercises 365

11 span {(1, 0) , (0, 1)} = R2 and 17 (a) If x, y ∈ U and x, y ∈ V , 1 −2 x, y ∈ U ∩ V .Thencx ∈ U and cx ∈ V x = b always has solution 01 so cx ∈ U ∩ V ,andx + y ∈ U and since coeﬃcient matrix is invertible. So x + y ∈ V so x + y ∈ U ∩ V . (b) Let 2 span {(1, 0) , (−2, 1)} = R and spans u1 +v1, u2 +v2 ∈ U +V ,whereu1, u2 ∈ agree. U and v1, v2 ∈ V .Thencu1 ∈ U and cv1 ∈ V so c(u1 + v1)=cu1 + cv1 ∈

2 2 U + V , and similarly for sums. 13 Write ax + bx + c = c1 + c2x + c3x as matrix system Ac =(a, b, c) by equat- 19 Let A and B be n × n diagonal ma- ing coeﬃcients and see whether A is in- trices. Then cA is diagonal matrix and vertible, or use an ad hoc argument. A + B is diagonal matrix so the set of (a) Spans P2. (b) Does not span P2 diagonal matrices is closed under matrix (can’t get 1). (c) Spans P2. (d) Does not addition and scalar multiplication. span P2 (can’t get x). 20 (a) If A =[aij ], vec(A)= (a11,a21,a12,a22)soforA there ex- 15 u + w =(4, 0, 4) and v − w = ists only one vec(A). If vec(A)= (−2, 0, −2) so span {u + w, v − w} = (a11,a21,a12,a22), A =[aij ]sofor span {(1, 0, 1)}⊂span {u, v, w},since vec(A) there exists only one A.Thusvec u + v, v − w ∈ span {u, v, w}. u is not operation establishes a one-to-one cor- a multiple of (1, 0, 1), so spans are not respondence between matrices in V and equal. vectors in R4.

Section 3.3, Page 180

1 (a) none (b) (1, 2, 1), (2, 1, 1), (3, 3, 2) 15 (c) W = −2, polynomials are linearly (c) every vector redundant (d) none independent (d) W = 4, polynomials are linearly independent 3 (a) linearly independent (b) linearly independent, (c) every vector redundant 1 0 17 1 (d) linearly independent 0 3 2 1 −3 1 3 5 (a) 4 , 4 (b) 2 , 1, 2 (c) (b, a, c) 22 Assume vi = vj . Then there exists 1 − − 3 − (d) 2 i, 1 2 i ci = cj = 0 such that c1v1 + c2v2 + ···+ civi + ···+ cj vj + ···+ cnvn = 0. 7 (a) v =3u1 − u2 ∈ span {u1, u2}, (b) u1, u2,(1, 0, −1) 24 Start with a nontrivial linear combination of the functions that sums to 0 9 With the given information, {v , v } 1 2 and diﬀerentiate it. and {v1, v3} are possible minimal spanning sets. 26 Domain and range elements x and y are given in terms of old coordinates. Ex- 11 All values except c =0,2or− 7 3 press them in terms of new coordinates 13 e11 x , y (x = P x and y = P y .) 366 Solutions to Selected Exercises

Section 3.4, Page 190 − 3 1 2 | R 1 (a) 2 , 0, 3, 1 , 2 , 1, 0, 0 a + bx + cx a + b + c =0 , range (b) {(−4, 1)} (c) {(−3, 1, 1)} (d) {} onto but not one-to-one 11 ker T = span {v − v + v }, 3 (a) {(2, 4) , (0, 1)} (b) {(1, −1)} 1 2 3 range T = R2, T is onto but not one- (c) {(1, −2, 1) , (1, −1, 2)} to-one, hence not an isomorphism. (d) {(2, 4, 1) , (−1, −2, 1) , (0, 1, −1)} 15 Calculate T (0 + 0). 5 (a) {(2, −1, 0, 3) , (4, −2, 1, 3)} (b) {(1, 4)} (c) {(1, 1, 2) , (2, 1, 5)} 17 Use deﬁnition of isomorphism, The- − { − − − } orem 3.9 and for onto, solve c1x + c2(x (d) (2, 1, 0) , (4, 2, 1) , (1, 1, 1) 2 2 1) + c3x = a + bx + cx for ci’s. { } 2 2 1 7 (a) span (2, 2, 1) , 5 , 5 , 5 ,yes 19 Since A is nilpotent, there exists { } 1 1 m m (b) span (1, 1) , 2 , 2 ,no m such that A = 0 so det(A )= (det A)m =0anddetA =0. Also since 9 (a) kernel span {(1, 0, −1)}, A is nilpotent, by Exercise 17 of Sec- range span {(1, 1, 2) , (−2, 1, −1)}, tion 2.4, (I − A)−1 = I + A + A2 + ...+ not onto or one-to-one (b) kernel Am−1.

Section 3.5, Page 196

1 (a) None (b) Any subset 13 If c1,1e1,1+···+cn,nen,n =0,ca,b =0 of three vectors (c) Any two of for each a, b because ea,b istheonlyma- {(2, −3, 1) , (4, −2, −3) , (0, −4, 5)} and trix with a nonzero entry in the (a, b)th (1, 0, 0) position.

3w1 could replace v2. 14 The union of bases for U and V will work. The fact that if u + v = 0, u ∈ U, 5w could replace v or v , w could re- 1 2 3 2 v ∈ V ,thenu = v = 0, helps. place any of v1, v2, v3 and w1, w2 could replace any two of v1, v2, v3. 16 Dimension of the space is n(n+1)/2. 1 2 2 7 (0, 1, 1), (1, 0, 0), (0, 1, 0) is one choice 21 I,A,A2,...,An must be linearly among many. dependent since dim(Rn,n)=n2 .Exam- 9 (a) true (b) false (c) true (d) true ine a nontrivial linear combination sum- (e) true (f) true ming to zero. 12 Suppose not and form a nontrivial linear combination of w1, w2,...,wr, w. Could the coeﬃcient of w be nonzero?

Section 3.6, Page 206

1 Bases for row, column, and null 3 Bases by row and column al- spaces: {(1, 0, 3, 0, 2) , (0, 1, −2, 1, −1)}, gorithms: (a) {(1, 0, 1) , (0, 1, −1)}, {(3, 1, 2) , (5, 2, 3)}, {(−3, 2, 1, 0, 0) , {(0,−1, 1) , (2, 1, 1)} − − } 1 (0, 1, 0, 1, 0) , ( 2, 1, 0, 0, 1) (b) 1, 0, 2 , (0, 1, 0) , {(2, −1, 1) , (2, 0, 1))} (c) {(1, 0) , (0, 1)}, Solutions to Selected Exercises 367 { − } 2 − 2 − (1, 1) , (2, 2) (d) 1+x ,x 5x , where c1 =2c3, c2 = c3,and c3 is 2 2 1+x2, −2 − x +3x2 free, dim span x, x + x, x − x =2 (c) c1v1 + c2v2 + c3v3 + c4v4 = 0,where 5 Bases for row, column, and null spaces: 1 c1 = −c3, c2 = c3, c4 = 0 and c3 is free, (a) {(2, 0, −1)}, {1}, 1 , 0, 1 , (0, 1, 0) 2 2 dim span {v1, v2, v3, v4} =3 (b) {(1, 2, 0, 0, 1) , (0, 0, 1, 1, 0)}, {(1, 1, 3), (0, 1, 2)}, {(−2, 1, 0, 0, 0) , 9 (a) dimC(A) = 2, dim C(B)=2 (0, 0, −1, 1, 0) , (−1, 0, 0, 0, 1)} (b) dim C AB =3(c)dimC(A) ∩ (c) {(1, 0, −10, 8, 0) , (0, 1, 5, −2, 0) , C(B)=2+2− 3=1 , , , , } (0 0 0 0 1) , 11 C(A) ∩C(B) = span {(1, 1, −1)} {(1, 1, 2, 2) , (2, 3, 3, 4) , (0, 1, 0, 1)}, {(10, −5, 1, 0, 0) , (−8, 2, 0, 1, 0)} 13 Since Ax = b is a consistent, b ∈ (d) {e1, e2, e3}, {e1, e2, e3}, {} C(A). If {ai}, the set of columns of A, has redundant vectors in it, c1a1 +c2a2 + 7 (a) c1v1+c2v2+c3v3+c4v4 = 0,where ···+ cnan = 0 for some nontrivial c. c1 = −2c3 − 2c4, c2 = −c3,andc3,c4 are free, dim span {v1, v2,v3, v4}=2 15 What does b ∈C/ (A) tell you about 2 2 (b) c1x + c2 x + x + c3 x − x =0 r and m?

Section 4.1, Page 219 √ √ √ √ 1 +1 1 (a)√−14,√ 34, 2 5(b)7,√ 6, √14 11 u = 2 , n2 → 0, 1 n n 2+ 3 + 5 2 (c) 8, √ 10, √ 26 (d) 12 √− 6i,√ 10, 26 n n2 − (e) 4, 30, 6(f) 4, 2 3, 30 ∈ Rn √ √ 13 Let u =(u1,...,un) , v = ∈ Rn ∈ R 3 √(a) − 145/145 (b) 0 (c) 21/6 (v1,...,vn) ,andc .Then (d) 10/10 (cu) · v =(cu1)v1 + ··· +(cun)vn and v · (cu)=v1(cu1)+··· + vn(cun)so 5 (a) 36k (b) −5i−j+5k (c) (−2, −2, 4) · · · √ √ (cu) v = v (cu). Similarly, show (cu) · · · 7 u = 30,√ cu =3 30,v = v = v (cu)=c(v u)=c(u v). 4,u + v = 30, u + v≤u + √ | | v =4+ 30 17 cv = c v by basic norm law (2). Since c ∈ R and c>0, cv = c v. 9u×v =(−6, 4, −8), v ×u =(6, −4, 8), So a unit vector in direction of cv is (cu) × v = c (u × v)=u × (cv)= cv/c v = v/ v. (12, −8, 16), u × w =(4, 1, −2), v × w = − (6, 7, 8) u × (v + w)=(−2, 5, −10), 19 Apply the triangle inequality to (u + v) × w = − (2, 6, 10) u +(v − u)andv +(u − v).

Section 4.2, Page 230

1 (a) 2.1176 (b) 1.6383 (c) 1.0018 7 (a) (Mu) · (Mv)=1,no(b)(Mu) · (Mv) = 0, yes (c) (Mu) · (Mv)=−13, √ − − − 10 10 no 3 (a) ( 2, 1), 5(b) 9 (2, 2, 1), 3 (c) −1 (1, 1, 1, 1), −1 2 9 (a) x + y − 4z = −6(b)x − 2z = −4 √ | · | ≤ − 2 − 5 (a) v1 v2 =1 v1 v2 = √15 11 (a) x = 3, 3 , b Ax = (b) |v1 · v2| =0≤v1v2 √= 6 0, b − Ax =0,yes(b)x = | · | ≤ 1 − − 1 − − (c) v1 v2 =19 v1 v2 =2 165 21 (9, 14), b Ax = 21 ( 4, 16, 8), 368 Solutions to Selected Exercises

√ − 336 b Ax = 21 ,no(c) x = 15 Express each norm in terms of dot 12 − 23 products. x3 + 13 , x3 + 26 ,x3 where x3 is free, b−Ax = 1 (32, −21, 1, 22), b − Ax = √ 26 21 Use Example 3.41. 1950 ,no 26 23 Examine the proof of Theorem 4.3 13 b =0.3, a + b =1.1, 2a + b =2, for points where real and complex dots 3a + b =3.5, 3.5a + b =3.6, least squares might diﬀer. solution a ≈ 1.00610, b ≈ 0.18841, residual norm is b − Ax≈0.39962

Section 4.3, Page 240 1 (a) orthogonal, linearly independent 12−2 1 (b) linearly independent (c) orthonor- 7 Hv = 3 21 2 , Hvu =(3, 0, 0), mal, orthogonal, linearly independent −22 1 Hvw =(1, 2, −2) 3v1 · v2 =0,v1 · v3 =0,and ⎡ √ √ √ ⎤ · { } 6 − 3 2 v2 v3 =0so v1, v2, v3 is an orthog- √6 √3 2 − ⎣ 6 3 ⎦ 1 3 4 onal basis. v1 · v1 =2,v2 · v2 =3, 9 (a) (b) √3 √3 √0 5 3 43 and v3 · v3 = . Coordinates with re- − 6 3 2 2 6 3 2 { } 3 −1 −5 spect to v1,v2, v3 are (a) 2 , 3 , 3 i0 1 −1 1 1 −5 11 (c) (b) , , (c) , , 01 2 3 3 2 3 3 34 11 Calculate both sides of each equa- 5 (a) orthogonal, 1 (b) not or- 5 4 −3 tion. thogonal (c) not orthogonal (d) not or- ⎡ ⎤ ,14 Let, u, vbe columns of P ,calculate − 10√ i ,eiθu, and eiθu · eiθv . thogonal (e) unitary, √1 ⎣ 0 − 2i 0 ⎦ 2 10i 1 − i −i (f) unitary, √1 3 −i1+i Section 4.4, Page 246 120 76−6 5 1 1 1 −10 , range (T )=25 18−8

011 span {(1, 1, 0), (2, −1, 1), (0, 0, 1)}, 10 Let B be any other basis and use the idV T idV ker(T )={0} chain of operators V → VB → VB → B VB . 11 23 3 (a) P = − , Q = 1 1 01 −1 −12 (b) [id]B,B = Q I2P = − 1 1 3 5 (c) [w] = [id] = B B,B 4 −1 Solutions to Selected Exercises 369

Section 5.1, Page 261

1 (a) −3, 2 (b) −1, −1, −1 (c) 2, 2, 3 11 (a) No (b) No (c) No (d) Yes (e) No (d) −2, 2 (e) −2i, 2i 13 Eigenvalues of A√are 1, 2. Eigenval- 3 Eigenvalue, algebraic multiplicity, geo- ues of B are 1 3 ± 5 . Eigenvalues of − 2√ metric multiplicity, bases: (a) λ = 3, A + B are 3 ± 3. Eigenvalues of AB are { } { } √ √ 1, 1, (2, 1) , λ =2,1,1, (1, 1) 3 ± 7. (a) Deny – 3 + 3 not sum of 1 − { } √ √ (b) λ = 1, 3, 1, (0, 0, 1) ,(c)λ =2, or 2 plus 1 3 ± 5 .(b)Deny–3+ 7 2, 2, {(1, 0, 0) , (0, −1, 1)}, λ =3,1,1, 2 √ not product of 1 or 2 times 1 3 ± 5 . {(1, 1, 0)} (d) λ = −2, 1, 1, {(−1, 1)}, 2 { } − λ =2,1,1, (1, 1) (e) λ = 2i, 1, 1, 17 If A is invertible, λ =0,then { − } { } − − (i, 1) , λ = 2i, 1, 1, (i, 1) A 1Av = A 1λv. 5 B =3I − 5A, so eigensystem for 19 For λ eigenvalue of A with eigenvec- B consists of eigenpairs {−2, (1, 1)} and tor v,(I − A)v = Iv − Av = v − λv = {8, (1, −1)}. (1 − λ)v.Since|λ| < 1, 1 − λ>0. 7 (a) tr A =7− 8=−1=−3+2, (b) tr A = −1 − 1 − 1=−3, (c) tr A = 20 Use part (1) of Theorem 5.1. 7 = 2+2+3 (d) trA =0+0=0=−1+1 22 Deal with the 0 eigenvalue separately. (e) tr A =0+0=0=−2i + 2i If λ is an eigenvalue of AB, multiply the 9 Eigenvalues of A and AT are the same. equation ABx = λx on the left by B. Section 5.2, Page 270 −11 −31 1 All except (d) have distinct eigenval- 9 P = , Q = , S = ues, so are diagonalizable. For λ =1(d) 11 20 has eigenspace of dimension two, so is not 11 −1 2 −1 , S = − diagonalizable. 12 11 110 12 0 √ √ − 1 3 4 + 2 3 3 (a) 001 (b) 00 1 11 sin π A = 2 5 5 , 6 0 −1 010 ⎡ 01 1⎤ 1 2 1 −11−1 cos π A = 2 5 2 −1 −11 ⎢−210−1⎥ 6 00 (c) 3 (d) (e) ⎣ ⎦ 11 11 0 −10 3 13 Similar matrices have the same eigen- 0200 values. 5 True in every case. (a), (b), and (c) 15 Examine DB = BD,withD diago- satisfy p (A) = 0 and are diagonalizable, nal and no repeated diagonal entries. (d) is not diagonalizable and p (A) =0.

7 Eλ (Jλ (2)) = span {(1, 0)},soJλ (2) 17 You may ﬁnd Exercise 16 and Corol- is not diagonalizable (not enough eigen- lary 5.1 helpful. 2 √ √ 2 λ 2λ 3 1+ 5 n 5+ 5 vectors). J (2) = , J (2) = 19 (b) fn =( ) ( )+ λ 0 λ2 λ √ √ 2 10 ( 1− 5 )n( 5− 5 ). 3 2 4 3 2 10 λ 3λ 4 λ 4λ 3 , Jλ (2) = 4 ,which 21 In one direction use the fact that di- 0 λ 0 λ k k−1 agonal matrices commute. In the other k λ kλ suggests Jλ (2) = . direction, prove it for a diagonal A ﬁrst, 0 λk then use the diagonalization theorem. 370 Solutions to Selected Exercises

Section 5.3, Page 280

1 (a) 2, dominant eigenvalue 2 (b) 0, no 9 Characteristic polynomial for J3 (2) − 3 − 3 dominant eigenvalue (c) 0, no dominant is (λ 2) and (J3 (2) 2I3) = eigenvalue (d) 1, dominant eigenvalue −1 010 3 1 −1 (e) 2 , dominant eigenvalue 2 001 =0. k k 000 2 1 − −1 (k) 2 2 3 (a) x = k k −2 1 +2 −1 11 Eigenvalues are λ ≈ 0.1636 ± ⎡ 2 ⎤ 2 2k 0.3393i, 0.973 with absolute values (b) x(k) = ⎣ 3k+1 − 2k ⎦ (c) x(k) = 0.3766 and 0.973. So population will decline at rate of approximately 2.7% 2k pertimeperiod. 13 · 2k − 10 · 3k − · k · k 13 2 +15 3 2 13 λ = s1f2 , p = p1 1, s1/f2 5 (b), (c), and (e) give matrices for which all x(k) → 0 as k →∞.Ergodic 15 (a) Sum of each column is 1. (c) theorem does not apply to any of these. Since a and b are nonnegative, (a, b)and 7 diag {A, B}, where possibili- (1, −1) are linearly independent eigen- ties for A are diag {J2 (1) ,J2 (1)}, vectors. Use diagonalization theorem. J2 (2) and possibilities for B are diag {J3 (1) ,J3 (1) ,J3 (1)}, 17 Show that (1, 1,...,1) is a left eigen- diag {J3 (1) ,J3 (2)},diag{J3 (3)} vector. Section 5.4, Page 286 ⎡ √ √ ⎤ T − 3 6 1 A is real and A = A √3 √3 √0 ⎣ 3 6 2 ⎦ 1 21 7 Orthogonalize by − , in each case. (a) √ √3 √6 √2 5 − ⎡ 12⎤ 3 6 2 3 6 2 −101 − k k+1 − k−1 k −43 let a =( 1) +2 , b =( 1) +2 , (b) 1 (c) √1 ⎣ 101⎦ 5 34 2 √ abb 0 20 − k k−1 k 1 ⎡ √ √ √ ⎤ c =( 1) +2 and A = 3 bcc − 2 6 3 bcc √2 √6 √3 (d) ⎣ 2 6 3 ⎦ 2 √6 √3 6 3 ⎡ √ √ √ ⎤ − 3 2 − 6 0 3 3 − √3 2 √6 T 3 P P = I in each case.⎡ (a) Unitar-⎤ 9 P = ⎣ 3 − 6 ⎦, √3 √0 √3 0 −ii 3 2 6 1 ⎣ ⎦ 3 √2 6 ily diagonalizable by √ 01 1 T 2 √ B = P diag 1, 2, 4 P 20−1 ⎡ √ √ ⎤ 2 + 2 1 −2 + 2 (b) Unitarily diagonalizable by 3 2 3 3 2 = ⎣ 1 5 −1 ⎦ −ii √3 3 √3 √1 (c) Orthogonally diagonal- −2 + 2 −1 2 + 2 2 11 ⎡ ⎤ 3 2 3 3 2 −101 izable by √1 ⎣ 101⎦ 12 Use orthogonal diagonalization and 2 √ 0 20 change of variable x = P y for a general B to reduce the problem to one of a di- 5 All of these matrices are normal. agonal matrix. Solutions to Selected Exercises 371

16 First show it for a diagonal matrix 17 AT A is symmetric. Now calculate with positive diagonal entries. Then use Ax2 for an eigenvector x. Exercise 12 and the principal axes theorem. Section 5.5, Page 287 −3005 Eigenvalues of A are 2, −3 and eigen- 1 (a) 0 −2.5764 −1.5370 values of f (A) /g (A)are0.6, 0.8. − 0 1.5370 2.5764 1.41421 0 0 8 Do a change of variables x = P y, (b) 0 −1.25708 0.44444i where P upper triangularizes A. 0 −0.44444i −0.15713 11 Equate (1, 1)th coeﬃcients of the equation R∗R = RR∗ and see what can √ be gained from it. Proceed to the (2, 2)th 3 (a) −2, 3, 2(b)3, 1, 2(c)2, −1, ± 2 coeﬃcient, etc. Section 5.6, Page 291 300 3 Calculate U, Σ, V , null space, column 1 (a) U = E2(−1), Σ = , ⎡ 010⎤ space bases: (a) First three columns of − {} 10√ √ 0 U, (b) First two columns of U,third ⎣ 2 − 2 ⎦ column of V V = I3 (b) U = 0 √2 √2 , 2 2 ⎡ ⎤ 0 2 2 5 For (3), use a change of variables 20√ x = V y. ⎣ 0 2 ⎦, V = I (c) U = E E , 2 12 23 7 Use a change of variables x = 00 ⎡ √ ⎤ ⎡ ⎤ V, y and check, , that b − Ax , = , T , , T T , 500 √01√ 0 U (b − Ax) = U b − U AV y . ⎣ ⎦ ⎣ 5 2 5 ⎦ Σ = 010 , V = √5 0 √5 000 −2 5 5 5 0 5 200 (d) U = E E (−1), , V = I 12 2 020 3 Section 5.7, Page 294

1 (a) −0.04283, 5.08996, 2.97644 ± 3 Use Gershgorin to show that 0 is not 0.5603 (b) −0.48119, 3.17009, 1.3111 an eigenvalue of the matrix. (c) 3.3123 ± 2.8466i, 1.6877 ± 0.8466i Section 6.1, Page 311 √ √ 1 (a) 1-norms 6, 5, 2-norms 14, 11, 3 (a) 1 (1, −3, −1), √1 (1, −3, −1), 5 11 ∞-norms 3, 3, distance ((−5, 0, −4)) 1 − − 1 − √ 3 (1, 3, 1) (b) 7 (3, 1, 1, 2), in each norm√ 9,√ 41, 5 (b) 1-norms 7, √1 (3, 1, −1, 2), 1 (3, 1, −1, 2) ∞ 15 3 8, 2-norms 15, 14, -norms 3, 2, dis- (c) √1 (2, 1, 3+i), √1 (2, 1, 3+i), tance√ ((1, 4, −1, −2, −5)) in each norm 3+ 10 15 √1 (2, 1, 3+i) 13, 47, 5 10 372 Solutions to Selected Exercises

5 u 1 =6,v 1 =7,are between the pairs of planes (1) x =0, (1) u 1 > 0, v 1 > 0(2) x =2,(2)y =0,y = 2 and (3) z =0, − |− | 2(0, 2, 3, 1) 1 =12= 2 6(3) z =2. (0, 2, 3, 1) + (1, −3, 2, −1) =7≤ 6+7 1 11 Set v =( −1, 1), v − −1 − −n − 7 Ball of radius 7/4 touches the line, so v n = n , e so v vn 1 = 1 −n distance from point to line in ∞-norm is + e −−−→−−−→ ∞ 0andv − v = n n n 2 7/4. ( 1 )2 +(e−n)2 → 0, as n →∞.So y n limn→∞ vn is the same in both norms. 3 13 Answer: max{| {|a| + |b| , |c| + |d|}. Note that a vector of length one has one 2 coordinate equal to ±1 and the other at (3/4,5/4) 1 most 1 in absolute value.

(0,0) 14 Let u =(u1,...,un), v = x (v1,...,vn), so |u1| + ···+ |un|≥0. Also 1 7/4 1 2 3 (1,1/2) | | ··· | | | || | ··· | || | + =2 cu1 + + cun = c u1 + + c un 1 x y | | ··· | |≤| | 7/4 and u1 + v1 + + un + vn u1 + ···+ |un| + |v1| + ···+ |vn|. 15 Observation that A F = vec (A) 2 3 9 Unit ball B1 ((1, 1, 1)) in R with in- enables you to use known properties of ﬁnity norm is set of points (x, y, z)which the 2-norm.

Section 6.2, Page 320 √ | | 2255 486 1129 ∈ 1 (a) √u, v = 46,√ u√ = 97, V (b) 437 , 437 , 437 ,(5, 1, 3) / V v = 40 and 46 ≤ 94 40 ≈ 61.32 (c) (5, 2, 3), (5, 2, 3) ∈ V (b) |u, v| = 1 , u = √1 , v = √1 5 3 7 T 13 vi Avj =0for i = j.Coordi- 1 ≤ √1 √1 ≈ and 5 =0.2 0.2182 7 5 1 1 1 3 7 nate vectors: (a) 2 , 6 , 3 (b) 0, 3 , 3 (c) (1, 1, 0) 3 projvu, comp v u,orth v u: (a) −23 , 23 , √46 , 63 , 7 (b) 7 x2, 15 ac + 1 (ad + bc)+ 1 bd √ 20 10 40 20 10 5 2 3 7 − 7 3 5 , x 5 x 17 Express u and v intermsofthestan- dard basis e1, e2 and calculate u, v. 5 If x =(x, y, z), equation is 4x − 2y + 2z =2. 18 Use the same technique as in Exam- ple 6.13. 7 Only (1), since if, e.g., x =(0, 1), then 19 Follow Example 6.8 and use the fact x, x = −2 < 0. ∗ that Au2 =(Au) Au. 9 (a) orthogonal (b) not orthogonal or 20 (1) Calculate u, 0 + 0. (2) Use orthonormal (c) orthonormal norm law (2), (3) and (2) on u + v, w.

2 2 11 1(−4) + 2 · 3 · 1+2(−1) = 22 Express u + v and u − v in v1,v terms of inner products and add. 0. For each v calculate v1 + v1,v1 v2,v v2. (a) (11, 7, 8), (11, 7, 8) ∈ 23 Imitate the steps of Example 6.9. v2,v2 Solutions to Selected Exercises 373

Section 6.3, Page 331 − 1 − 1 − 1 −1 1 (a) (1, 1, 1), 3 ( 2, 7, 5), 13 (8, 2, 6) v2 =(1, 1, 1, 1), v3 = 2 , 0, 0, 2 , v4 = 1 − −1 1 (b) (1, 0, 1), 2 (1, 8, 1) (c) (1, 1), 0, , , 0 . 1 − − − 2 2 2 ( 1, 1) (d) (1, 1, 1, 1), (0, 1, 1, 0), 1 − 11 Use Gram–Schmidt on columns 4 (5, 1, 1, 3) of A and normalize to obtain ortho- 100 normal √1 (1, 1, 1) and √1 (1, 2, −5), 1 −1 3 42 3 (a) 1 (b) 010 2 −11 563 001 1 − ⎡ ⎤ then projection matrix 14 610 2 . 14 1 −23 − 524 3 213 ⎢ 1142−3⎥ (c) 1 ⎣ ⎦ (d) 1 28−2 Use Gram–Schmidt on columns of B 15 −2 2 11 6 9 4 −25 and normalize to obtain orthonormal 3 −36 6 √1 (4, 5, 2), √1 (−1, 10, −23), then ob- 3 5 3 70 1 − tain same projection matrix. 5 projV w,orthV w:(a) 6 (23, 5, 14), 1 − − 1 1 − 3 6 (1, 1, 2) (b) 3 (4, 2, 1), 3 ( 1, 1, 2) 14 If a vector x ∈ R is projected into 1 − 1 − 3 (c) 3 (1, 1, 1), 3 ( 1, 1, 2) R , the result is x. 3 1 − 7, projV x , = 10 (9x 2), 16 Use matrix arithmetic to calculate ,x3 − 1 (9x − 2), = 3√ P u, v − P v. 10 10 7 9 Use Gram–Schmidt algorithm on 18 For any v ∈ V , write b − v = w1 =(−1, 1, 1, −1), w2 =(1, 1, 1, 1), (b − p)+(p − v), note that b − p is or- w3 =(1, 0, 0, 0), w4 =(0, 0, 1, 0) to ob- thogonal to p − v, which belongs to V , tain orthogonal basis v1 =(−1, 1, 1, −1), and take norms.

Section 6.4, Page 341 ⎡ ⎤ ⊥ 1 5 −1 −1 √1 √2 1 V = span 2 , 2 , 1, 0 , 2 , 2 , 0, 1 5 6 ⎣ 0 √1 ⎦ and if A consists of the columns trix is rank deﬁcient. 6 , − − 1 , 5 , 1, 0 , 1 , 1 , 0, 1 ,(1, −1, 2, 0), √−2 √1 2 2 2 2 5 6 (2, 0, −1, 1), then det A =18which √ − 50 −√10 2x3 3 shows that the columns of A are linearly √ 5 − 12 , 2x3 +2 , x3 free independent, hence a basis of R4. 0 6 √ ⎡ 6 ⎤ x3 5 ⎡ ⎤ 10√ 3 3 ⊥ 3 38 2 20 3 V = span − x + x ⎢ 1 2 −1 ⎥ √ √2 14 35 (c) 1 ⎣ √ 3 ⎦, ⎣ 0 2 3 2 ⎦, 2 −1 2 1 2 3 00 3 ⊥ − − 2 5 V = span −2, 1 , 1 and 101 2 − ⊥ 1 ⊥ 1 −1 2 V = span 2 , 0, 1 , 4 , 1, 0 9 1 2 −5 which is V since (1, 0, 2) = 2 2 ,0, 1 and 1 −1 3 (0, 2, 1) = 2 , 0, 1 +2 4 , 1, 0 . 11 (a) Inclusion U ⊥ + V ⊥ ⊂ (U ∩ ⎡ ⎤ V )⊥follows from the deﬁnition and in- 3 √4 5 ∩ ⊂ 5 2 52 clusion U V U +V . For the converse, 7 Q, R, x:(a)⎣ 0 √1 ⎦, √ , − 2 0 2 show that (v projU v) is orthogonal to 4 −√3 ∈ ⊥ ⊥ 5 5 2 all u U.(b)Use(a)onU , V . 9 5 T −5 (b) Caution: this ma- 12 Show that if A Ay = 0,thenAy = 2 0. 374 Solutions to Selected Exercises

Section 6.5, Page 347

1 Frobenius,√ 1-, and ∞√-norms: √ 5 Use the triangle inequality on A and −1 (a)⎡ 14, 3, 5 (b) 3⎤ 3, 5, 5 (c) 2 17, and Banach lemma on A . 12 2 1 6 Factor out A and use Banach lemma. ⎢ 1 −30−1 ⎥ 10, 9 ⎣ ⎦ 11−20 10 Examine Ax with x an eigenvector −21 6 0 belonging to λ with ρ (A)=|λ| and use deﬁnition of matrix norm. 3 Verify that the perturbation theorem 120 −5 11 If eigenvalue λ satisﬁes |λ| > 1, con- is valid for A = 01−2 , b = 1 , sider Amx with x an eigenvector be- 0 −21 −3 longing to λ. For the rest, use the Jordan δA =0, .05A,and, δb = −0.1b.Calculate canonical form theorem. , −1 , c = A δA =0.05 I3 =0.05 < 1, 13 (a) Make change of variables x = V y δA =0.05, δb =0.05, cond (A) ≈ , , A b " # and note ,U T AV x, = Ay , x = 2 2 cond(A) δA δb ≈ 6.7807. Hence, 1−c A + b V y . (c) Use SVD of A. δx 1 0.71376. Now calculate x = 7 and 1 ≈ 7 0.142 < 0.71376. Section 6.6, Page 354 ⎡ √ ⎤ − 1x=(0.4, 0.7), δx∞ / x∞ = √ 3 24 4 2 ⎣ ⎦ 1.6214, cond (A) δb ∞ / b ∞ =1.8965 3 Q = 10 √055, R = − ⎡ ⎤ 4 2 33 52√ ⎣ ⎦ 1 − 0 2 , x = 10 (4, 5), b Ax 2 = 00 9 2 References

1. Ake˙ Björck. Numerical Methods for Least Squares Problems.SIAM, Philadelphia, PA, 1996. 2. Tomas Akenine-Möller and Eric Haines. Real-Time Rendering. A K Peters, Ltd, Natick, MA, 2002. 3. Richard Bellman. Introduction to Matrix Analysis. SIAM, Philadelphia, PA, 1997. 4. Hal Caswell. Matrix Population Models. Sinaur Associates, Sunderland, MA, 2001. 5. G. Caughley. Parameters for seasonally breeding populations. Ecology, 48:834– 839, 1967. 6. Biswa Nath Datta. Numerical Linear Algebra and Applications. Brooks/Cole, New York, 1995. 7. James W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997. 8. C. F. Gauss. Theory of the Combination of Observations Least Subject to Errors, Part 1. Part 2, Supplement, G. W. Stewart. SIAM, Philadelphia, PA, 1995. 9. G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins Uni- versity Press, Baltimore, Maryland, 1983. 10. Per Christian Hansen. Rank-Deficient and Discrete Ill-Posed Problems.SIAM, Philadelphia, PA, 1998. 11. R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, UK, 1985. 12. P. Lancaster and M. Tismenetsky. The Theory of Matrices. Academic Press, Orlando, Florida, 1985. 13. Lloyd Trefethen and David Bau. Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997. Index

Abstract vector space 150 Column space 183 Adjacency matrix 81 Companion matrix 128, 299 Adjoint Complex number 12 formula 123 argument 16 matrix 122 Euler’s formula 15 Affine set 200 imaginary part 12 Angle 222, 316 Polar form 16 Argument 16 real part 12 Augmented matrix 24 Complex plane 12 Component 224, 319 Ball 308 Condition number 344, 348 closed 308 Conditions for matrix inverse 108 open 308 Conductivity 53 Banach lemma 344 Conformable matrices 63 Basis 173 Conjugate symmetric 94 coordinates relative to 176 Consistency 29 ordered 174 in terms of column space 199 Basis theorem 192 in terms of rank 40 Block matrix 90, 98, 114 Coordinate change 178 Bound variable 27 Coordinates 175 CAS 7 orthogonal 234, 319 Cayley–Hamilton theorem 198, 271, standard 176 280 vector 176 CBS inequality 221, 315 Counteridentity 128 Change of basis 179, 182, 242, 244 Cramer’s rule 124 Change of basis matrix 179, 243 Cross product 218 Change of coordinates 178, 182, 241 Change of variables 177, 178, 241 de Moivre’s Formula 16 Characteristic Determinant 115 equation 253 computational efficiency 132 polynomial 253 proofs of laws 132 Coefficient matrix 24 Diagonal 89 Cofactors 117 Diagonalizable 378 Index

matrix 268 roundoff 46 orthogonally 282 Euler’s formula 15 unitarily 282, 289 Diagonalization theorem 268 Factorization Diagonalizing matrix 268 full QR 351 Diffusion process 4, 138 LU 129 Digraph 80 QR 338, 339 adjacency matrix 81 Fibonacci numbers 271 walk 80 Fieldofscalars 145 weighted 210 Finite-dimensional 192 Dimension 177 Fixed-point 85 definition 194 Flat 200 theorem 194 Flop 48 Direction 213 flop Discrete dynamical system 76 count 48 Displacement vector 147 Fourier analysis 353 Domain 156, 189 Fourier coefficients 353 Dominant eigenvalue 295 Fourier heat law 53 Dot product 217 Fredholm alternative 109, 337 Free variable 27 Eigenpair 251 Frobenius norm 310 Eigenspace 254 Full column rank 40 Eigensystem 254 Full row rank 40 algorithm 255 Function tridiagonal matrix 304 continuous 151, 156, 159 Eigenvalue 251 linear 71 dominant 295, 303 Fundamental Theorem of Algebra 14 repeated 260 simple 260 Gauss–Jordan elimination 25, 33 Eigenvector 251 Gaussian elimination 33 left 252 complexity 49 right 252 Gershgorin circle theorem 295 Elementary Givens matrix 178 column operations 93 Gram–Schmidt algorithm 323 inverse operations 35 Graph 79, 81, 142 matrices 86 adjacency matrix 81 row operations 24 dominance-directed 79, 81 transposes of matrix 93 loop 208 Elementary matrix walk 81 determinant 119 Equation Heat flow 4, 51–53 linear 3 Hermitian matrix 94 Sylvester 136 Householder matrix 237, 241, 288, 350 Equivalent linear system 35 Hyperplane 225 Equivalent norms 346 Ergodic Idempotent matrix 70, 100, 329 matrix 185, 279 Identity 160 theorem 279 Identity function 156 Error Identity matrix 65 Index 379

Image 189 trivial 171, 187 Imaginary part 12 zero value 171 Induced norm 315 Linear dependence 171 Inner product 92, 217 Linear function 71 abstract 312 Linear independence 171 Sobolev 323 Linear system space 312 coefficient matrix 24 standard 216, 314 equivalent 35 weighted 313 form of general solution 200 Input–output right-hand-side vector 24 matrix 7 List 170 model 6, 8, 9 Loop 80, 208 Integers 11 LU factorization 129 Interpolation 9 Intersection 169, 333 Markov chain 76, 77 set 10 MAS 7 Inverse 101, 123 Matrix Inverse iteration method 297 adjoint 122 Inverse power method 298 block 90 Inverse theory 52, 139 change of basis 179, 243 Isomorphic vector spaces 189 cofactors 122 Isomorphism 189 companion 299 complex Householder 239 Jordan block 278 condition number 344 Jordan canonical form 278, 299 conjugate symmetric 94 defective 261 Kernel 188, 189 definition 22 Kirchhoff diagonal 89 first law 9, 210 diagonalizable 268 second law 210 diagonalizing 268 Kronecker delta 122 difference 56 Kronecker symbol 65 elementary 86, 93 entry 23 Leading entry 23 equality 55 Least squares 227, 330 ergodic 185, 279 solution 229 exponent 67 solver 341, 352 full column rank 40 Left eigenvector 252 full row rank 40 Legendre polynomial 324 Givens 178 Leontief input–output model 6 Hermitian 94 Leslie matrix 302 Hilbert 53 Limit vector 85, 186, 214, 220 Householder 237, 241 Linear idempotent 70, 329, 332 mapping 156 identity 65 operator 72, 156 inverse 123 regression 227 invertible 101 transformation 156 leading entry 23 Linear combination 57, 164 minors 122 nontrivial 171 multiplication 63 380 Index

negative 56 algebraic 260 nilpotent 70, 100 geometric 260 nonsingular 101 Multipliers 130 normal 99, 286, 289 of a linear operator 243 Natural number 10 operator 72 Network 9, 208 order 23 Newton orthogonal 236 method 110 permutation 131 formula 111 pivot 26 Nilpotent matrix 70, 100, 113, 191 positive definite 229 Nonsingular matrix 101 positive semidefinite 228 Norm power bounded 348 p-norm 306 projection 241, 329 complex 212 pseudoinverse 293 equivalent 309, 346 reflection 241 Frobenius 310, 342 rotation 75, 178 general 305 scalar 89 induced 315 scalar multiplication 57 infinity 306, 310 similar 247, 265 matrix 342 similarity transformation 265 operator 343 singular 101 standard 212 size 23 uniform 310 skew-symmetric 100, 168 Normal equations 228 square 23 Normal matrix 99, 286, 289 square root 287 Normalization 213, 220 standard 243 Normed space 306 stochastic 77 Notation strictly diagonally dominant 303 for elementary matrices 25 sum 56 Null space 184 superaugmented 106 Nullity 39 symmetric 94, 332 Number Toeplitz 294 complex 12 transformation 72 integer 11 transition 76 natural 10 triangular 89 rational 11 tridiagonal 89 real 11 unitary 236 Numerical linear algebra 46 Vandermonde 128 vectorizing 137 One-to-one 188 zero 58 function 156, 157 Matrix norm 342 Onto function 156, 157 infinity 348 Operator 156 Matrix, strictly triangular 89 additive 156 Max 40 domain 189 Min 40 fixed-point 85 Minors 117 identity 160 Monic polynomial 253 image 189 Multiplicity invertible 157 Index 381

kernel 188, 189 Principal axes theorem 284, 289 linear 72, 156 Probability distribution vector 77 one-to-one 156, 157, 188 Product onto 156, 157 inner 92 outative 157 outer 92 range 189 Projection 224, 319, 327 rotation 73 column space formula 330 scaling 73 formula 223, 319 standard matrix 243 formula for subspaces 327 target 189 matrix 329 zero 160 orthogonal 225 Order 23 orthononal 328 Order of matrix 23 parallel 224, 319 Orthogonal problem 327 complement 333 theorem 328 complements theorem 336 Projection formula 223, 319 coordinates theorem 234, 319 Projection matrix 241 matrix 236 Pythagorean theorem 231, 317, 320 projection formula 329 set 233, 319 QR algorithm 354 vectors 222, 316 QR factorization 338, 339 Orthogonal coordinates theorem 234 full 351 Orthogonal projection 225, 328 Quadratic form 94, 99, 100, 300 Orthonormal set 233, 319 Quadratic formula 15 Outer product 92 Quaternions 249

Parallel vectors 223 Range 189 Parallelogram law 318 Rank 39 Partial pivoting 47 full column 229 Perturbation theorem 345, 348 of matrix product 97 Pivot 26 theorem 205 strategy 47 Rational number 11 Pivoting Real numbers 11 complete 47 Real part 12 Polar form 16 Real-time rendering 73, 141 Polarization identity 322 Reduced row echelon form 36 Polynomial 15 Reduced row form 36 characteristic 253 Redundancy test 171 companion matrix 128 Redundant vector 170 Legendre 324 Reflection matrix 241 monic 253 Regression 227 Positive definite matrix 229, 232, 287, Residual 227 313 Right eigenvector 252 Positive semidefinite matrix 228, 232 Roots 15 Power of unity 15 matrix 67 theorem 15 vertex 80 Rotation 73 Power bounded matrix 348 Rotation matrix 178, 237 Power method 296 Roundoff error 46 382 Index

Row operations 24 Strictly diagonally dominant 303 Row space 184 Subspace deﬁnition 161 Scalar 21, 89, 145 intersection 169 Scaling 73 projection 327 Schur triangularization theorem 288 sum 169 Set 10, 170 test 161 closed 308 trivial 163 empty 10 Sum of subspaces 169, 333 equal 10 Superaugmented matrix 106 intersection 10, 169 Supremum 343 prescribe 10 SVD 291 proper 10 Symmetric matrix 94 subset 10 System union 10 consistent 29 Shearing 73 equivalent 33 Similar matrices 247, 265 homogeneous 41 Singular inconsistent 29 values 292 inhomogeneous 41 vectors 292 linear 4 Singular matrix 101 overdetermined 227 Singular Value Decomposition 291 Skew-symmetric 168 Target 156, 189 Skew-symmetric matrix 100, 128 Tensor product 136 Solution Toeplitz matrix 294 general form 28 Trace 262 genuine 229 Transform 73 least squares 229 aﬃne 141 non-unique 26 homogeneous 161 set 34 translation 161 to zn = d 17 Transformation 156 to linear system 3, 21 Transition matrix 76 vector 34 Transpose 91 Space rank 95 inner product 312 Triangular 89 normed 306 lower 89 Span 164 strictly 89, 100 Spanning set 166 unit 130 Spectral radius 273 upper 89, 117, 271, 290 Square matrix 23 Tridiagonal matrix 89, 262 Standard eigenvalues 304 basis 174 Tuple coordinates 175 convention 34 form 12 notation 34 inner product 216 norm 212 Unique reduced row echelon form 37 vector space 149 Unit vector 213 States 76 Unitary matrix 236 Steinitz substitution 193 Upper bound 343 Index 383

Vandermonde matrix 128, 352 quaternion 249 Variable redundant 170 bound 27 residual 227 free 27 solution 34 Vec operator 137 subtraction 146 Vector unit 213 angle between 222, 316 Vector space convergence 214 abstract 150 coordinates 176 finite-dimensional 192 cross product 218 geometrical 146 definition 22, 150 homogeneous 148 direction 213 infinite-dimensional 192 displacement 149 inner product 312 homogeneous 141 laws 150 limit 85, 186, 214, 220 normed 305 linearly dependent 171 of functions 151 linearly independent 171 of polynomials 153, 163 opposite directions 213 standard 147 orthogonal 222, 316 parallel 223 Walk 80, 81 product 63 Wronskian 181 Undergraduate Texts in Mathematics (continued from p.ii)

Lang: Undergraduate Analysis. Ross: Differential Equations: An Introduction with Laubenbacher/Pengelley: Mathematical Expeditions. Mathematica®. Second Edition. Lax/Burstein/Lax: Calculus with Applications and Ross: Elementary Analysis: The Theory of Calculus. Computing. Volume 1. Samuel: Projective Geometry. LeCuyer: College Mathematics with APL. Readings in Mathematics. Lidl/Pilz: Applied Abstract Algebra. Second edition. Saxe: Beginning Functional Analysis Logan: Applied Partial Differential Equations, Second Scharlau/Opolka: From Fermat to Minkowski. edition. Schiff: The Laplace Transform: Theory and Logan: A First Course in Differential Equations. Applications. Lovász/Pelikán/Vesztergombi: Discrete Mathematics. Sethuraman: Rings, Fields, and Vector Spaces: An Macki-Strauss: Introduction to Optimal Control Approach to Geometric Constructability. Theory. Shores: Applied Linear-Algebra and Matrix Analysis. Malitz: Introduction to Mathematical Logic. Sigler: Algebra. Marsden/Weinstein: Calculus I, II, III. Second edition. Silverman/Tate: Rational Points on Elliptic Curves. Martin: Counting: The Art of Enumerative Simmonds: A Brief on Tensor Analysis. Second edition. Combinatorics. Singer: Geometry: Plane and Fancy. Martin: The Foundations of Geometry and the Singer: Linearity, Symmetry, and Prediction in the Non-Euclidean Plane. Hydrogen Atom. Martin: Geometric Constructions. Singer/Thorpe: Lecture Notes on Elementary Topology Martin: Transformation Geometry: An Introduction to and Geometry. Symmetry. Smith: Linear Algebra. Third edition. Millman/Parker: Geometry: A Metric Approach with Smith: Primer of Modern Analysis. Second edition. Models. Second edition. Stanton/White: Constructive Combinatorics. Moschovakis: Notes on Set Theory. Second edition. Stillwell: Elements of Algebra: Geometry, Numbers, Owen: A First Course in the Mathematical Foundations Equations. of Thermodynamics. Stillwell: Elements of Number Theory. Palka: An Introduction to Complex Function Theory. Stillwell: The Four Pillars of Geometry. Pedrick: A First Course in Analysis. Stillwell: Mathematics and Its History. Second edition. Peressini/Sullivan/Uhl: The Mathematics of Nonlinear Stillwell: Numbers and Geometry. Programming. Readings in Mathematics. Prenowitz/Jantosciak: Join Geometries. Strayer: Linear Programming and Its Applications. Priestley: Calculus: A Liberal Art. Second edition. Toth: Glimpses of Algebra and Geometry. Second Protter/Morrey: A First Course in Real Analysis. Edition. Second edition. Readings in Mathematics. Protter/Morrey: Intermediate Calculus. Second edition. Troutman: Variational Calculus and Optimal Control. Pugh: Real Mathematical Analysis. Second edition. Roman: An Introduction to Coding and Information Valenza: Linear Algebra: An Introduction to Abstract Theory. Mathematics. Roman: Introduction to the Mathematics of Finance: Whyburn/Duda: Dynamic Topology. From Risk management to options Pricing. Wilson: Much Ado About Calculus.