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.