bibliography - springer978-1-4612-1136-5/1.pdf · bibliography albert, a. a. (ed.) ... birkhoff and...

Bibliography

Albert, A. A. (ed.), Studies in Mathematics, Vol. II: Studies in ModernAlgebra (Buffalo: Mathematical Association of America, 1963).

Artin, E., Geometric Algebra (New York: Interscience, 1957).Benson, C. T., and L. C. Grove, Finite Reflection Groups (Tarrytown-on

Hudson, N.¥.: Bogden and Quigley, 1971).Birkhoff, G., and S. MacLane, Survey of Modern Algebra, rev. ed. (New

York: Macmillan, 1953).Bourbaki, N., Algebre, Chapitre 2, "Algebre lineaire," 3rd ed. (Paris:

Hermann, Actualites et Industrielles, no. 1144).Boyce, W. E., and R. C. DiPrima, Elementary Differential Equations and

Boundary Value Problems (New York: John Wiley, 1969).Courant, R., and H. Robbins, What is Mathematics? (New York:

Oxford University Press, 1941).Feller, W., An Introduction to Probability Theory and its Applications,

Volume One (New York: John Wiley and Sons Inc., 1950.)Halmos, P. R., Finite Dimensional Vector Spaces, 2nd ed. (Princeton:

Van Nostrand, 1958).Jacobson, N., Lectures in Abstract Algebra, Vol. II: Linear Algebra

(Princeton: Van Nostrand, 1953).Kaplansky, I., Linear Algebra and Geometry (Boston: Allyn and Bacon,

1969).MacLane, S., and G. Birkhoff, Algebra (New York: Macmillan,

1967).Noble, B., Applied Linear Algebra (Englewood Cliffs, N.J.: Prentice-Hall,

1969).

325

326 BIBLIOGRAPHY

Noble, B., Applications of Undergraduate Mathematics in Engineering(New York: The Mathematical Association of America and TheMacmillan Company, 1967).

Polya, G., How to Solve It (Princeton: Princeton University Press, 1945).Schreier, 0., and E. Sperner, Modern Algebra and Matrix Theory, English

translation (New York: Chelsea, 1952).Smith, K. T., Primer of Modern Analysis (Tarrytown-on Hudson, N.¥.:

Bogden & Quigley, 1971).Synge, J. L., and B. A. Griffith, Principles of Mechanics (New York:

McGraw-Hili, 1949).Van der Waerden, B. L., Modern Algebra, Vols. I and II, English transla

tion (New York: Ungar, 1949 and 1950).Weyl, H., Symmetry (Princeton: Princeton University Press, 1952).Wielandt, H., Topics in the Analytic Theory of Matrices, Lecture Notes

prepared by Robert R. Meyer from a course by Helmut Wielandt(Madison, Wisconsin: Department of Mathematics, University ofWisconsin, 1967.)

Books which had an influence on the writing of the present one, andfrom which, in some cases, material was used as a basis for the approachtaken in this book, are those by Bourbaki, Feller, Halmos, Jacobson,Kaplansky, Noble, Schreier and Sperner, and Wielandt. The books ofBirkhoff and MacLane, MacLane and Birkhoff, and Van der Waerdencontain comprehensive accounts of the axiomatic systems of modernalgebra (groups, rings, fields, etc.) which have all appeared naturallyin this book. The books of Artin, Benson and Grove, Kaplansky, andWeyl are recommended for further discussion of the connections between linear algebra and geometry, while the books of Boyce andDiPrima, Noble, Smith, and Wielandt contain many illustrations of theusefulness of linear algebra in analysis.

Solutions ofSelected Exercises

In the case of numerical problems where a simple check is available, noanswers are supplied. On theoretical problems, comments or hints on onemethod of solution are given, but not all the details. Of course there areoften many valid ways to do a particular problem, and a correct solution maynot always agree with the one given below.

SECTION 2

1. (a) The statement holds for k = I. Assume it holds for k ;::: I. Then

1 + 3 + 5 + ... + (2k - I) + [2(k + I) - 1 1= k2 + 2(k + 1) - 1 = (k + 1)2.

2. (a) u = cx/fJ and v = y/8 are solutions of the equations fJu = cx and8v = y, respectively. Multiply the equations by 8 and fJ and add,obtaining fJ8(u + v) = 0:8 + fJy.

3. Suppose for some n,

Then

Collect terms and use the definition of (n Z1) to complete the proof.

327

328 SOLUTIONS OF SELECTED EXERCISES

SECTION 3

1. (a) <1,4, -2>.(b) <-2,4,2> + <-2, -1,3> + <0,1,0> = <-4,4,5>.(d) <-a + 2f3, 2a + f3 + Y, a - 3f3>.

2. Check your answers by substitution in the equations.~ ~ -:lrr.-:lrr.

6. SinceAB = B - A,andCD = D - C,AB = CD implies that B - A =

D - C. The vector X = C - A = D - B behaves as required.Conversely, if C = A + X, D = B + X, then C - D = (A + X)

--" --"

- (B + X) = A - B, from which AB = CD follows by multiplyingboth sides by - I.

7. (a) <I, t>. (b) <2, D.8. (a) and (c) are vertices of parallelograms; (b) is not.

SECTION 4

1. (a) Not a subspace; (b) subspace; (c) subspace; (d) not a subspace;(e) subspace; (f) this set is a subspace if and only if B = 0; (g) not asubspace.

3. (a) Subspace; (b) not a subspace; (c) subspace; (d) not a subspace;(e) subspace; (f) subspace; (g) subspace; (h) this set is a subspace if andonly if g is the zero function: g(x) = 0 for all x.

4. (a) Linearly independent; (b) linearly dependent, - 3<1, I> + (2, I> +(1,2> = 0; (c) linearly independent; (d) linearly dependent, f3(O, I>+ a<l, 0> - (a, f3> = 0; (e) linearly dependent, <I, 1,2> - (3, 1,2> 2( - I, 0, 0> = 0; (f) linearly independent; (g) linearly dependent.

5. {<I, I, 0), (0, I, I)} is one solution of the problem.

6. Let 11 , ... ,In be a finite set of polynomial functions. Let xn be thehighest power of x appearing with a nonzero coefficient in any of thepolynomials {It}. Then every linear combination of 11, ... .In has theform aD + alX + ... + anxn . But there certainly exist polynomials,such as x n + 1, which cannot be exptessed in this form. To be certainon this point, we observe that upon differentiating n + 1 times, all linearcombinations of 11, .. . .In become zero, while there are polynomialswhose (n + I)st derivative is different from zero.

7. Yes. Let Sand T be subspaces. Let a, b E S n T. Then a + bE Sand a + bET so a + bE S n T. Similarly, if a E S n T and a E F,aaESn T.

8. No. For example, in R2 , let S = S«I, 0», T = S«O, I»). Then<1,1) = <1,0) + (0, I>¢SUT.

SECTION 5

1. Suppose b1 = 0, for example. Then I . b1 + 0 . b2 + ... + 0 . br = 0,and the vectors {b1 , .•• , br} are linearly dependent.

SOLUTIONS OF SELECTED EXERCISES 329

4. If d//dr is identically zero, then / is a constant, and / = c/o, where /0 isthe function everywhere equal to 1. The dimension of the subspaceconsisting of all / whose second derivative is zero is two.

5. Suppose alai + ... + amGm = a;GI + ... + a~G",. Then (al - a;)al+ ... + (am - a~)am = O. By the linear independence of aJ , ... , am,

we have ((1 = a~, ... , am = a~.

SECTION 6

2. and 3. (a) Linearly dependent (check your relation of linear dependencein this and subsequent problems), basis: < - 1, I>, <0, 3>; (b)linearly dependent, basis <2, 1>, <0, 2>; (c) linearly dependent,basis <1,4,3>, <0, 3, 2>; (d) linearly dependent, basis <1,0,0>,<0, I, 0>, <0,0, 1>; (e) linearly dependent; (0 linearly dependent.

5. (a) linearly dependent: (b) linearly independent.

SECTION 7

1. It does not. The subspace spanned by <I, 3,4>, <4,0, I>, and <3, 1,2>has a basis <I, 3,4>, <0,4, 5>. An arbitrary linear combination of thesevectors has the form <a, 3a + 4f3,4a + 5f3>. If <I, I, I> is one of theselinear combinations, then a = I, 3 + 4f3 = I, f3 = - 1, and 4a + 5f3 =4-1#1.

2. It does. A basis for the subspace, in echelon form, is <I, - I, 1, 0>,<0,2, I, -I>, <0,0, -t, - 1). A typical element in the subspace is<a, -a + 2f3,a + f3 -!Y, -f3 - h>. Comparingwith<2,0, -4, -2>,we obtain a = 2, f3 = I, y = 2.

3. Let (tl, ... , rml be a basis for T, and let SeT. Then every set ofm + 1 vectors in S is linearly dependent, by Theorem (5.1). Let(51, ... , sd be a set of linearly independent vectors in S such that everyset of k + 1 vectors is linearly dependent. By Lemma (7.1), everyvector in S is a linear combination of{sl, ... , Sk). Therefore {SI' ... , Sk}is a basis for S, and dim S S; dim T. Finally, let dim S = dim T andSeT. Let {SI, ... , sm) be a basis for S. Then by Theorem (5.1),{51, ... , S"', t} is linearly dependent for all rET, since dim T = m.By Lemma (7.1) again, rES and S = T.

4. dim (S + T) S; 3. Therefore dim (S n T) = dim S + dim T - dim(S + T) ~ 1, by Theorem (7.5).

5. dim S = dim T = 3, dim (S + T) = 4, dim (S n T) = 2.

6. There are 4 vectors in V, 3 one-dimensional subspaces, and 3 differentbases.


SECTION 8

1. (a) Solvable; (b) solvable; (c) solvable; (d) solvable; (e) solvable; (f)not solvable; (g) not solvable.

2. There is a solution if and only if a i= 1.

3. It is sufficient to prove that the column vectors of an m-by-n matrix, withn > m, are linearly dependent. The column vectors belong to Rm ,

and since there are n of them, they are linearly dependent by Theorem(5.1).

4. This result is also a consequence of Theorem (5.l).

SECTION 9

1. The dimension of the solution space is 2. Let Cl , C2, Ca, C4 be the columnvectors. Then Cl + 3C2 - 4ca = 0 and Cl + C2 - 2C4 = O. Thereforea basis for the solution space is <t, 3, - 4, 0> and <t, I, 0, - 2>.

2. The dimensions of the solution spaces are as follows. The actualsolutions should be checked:

(a) zero (b) one (c) two (d) one(e) two (f) one (g) zero

3. By Theorem (8.9), every solution has the form Xo + x where Xo is thesolution of the nonhomogeneous system and x is a solution of thehomogeneous system. The dimension of the solution space of thehomogeneous system is two, and the actual solutions found may bechecked by substitution.

S. A, B, and C must satisfy the equations

3A + B + C = 0-A + C = 0

or

A (_~) + B (~) + C G) = O.

The solution space of the system has dimension one, so that any twonontrivial solutions are multiples of each other.

6. We have to find a nontrivial solution to the system

A (~) + B (~) + C G) = O.

The rank of the matrix is at most two, so there certainly exists a nonzerosolution. To show that two such solutions are proportional, we have toshow that the rank is two. If the rank is one, then a = y and f3 = 8,and the points are not distinct, contrary to assumption.

SECTION 10

3. The result is immediate from Theorem (10.6).

4. Since L is one-dimensional, L = p + V where V is the directing space,and pEL. By (10.2) q - p E V, and since dim V = I, V consists of allscalar multiples of q - p.


5. This result follows from the definition of hyperplane in Exercise 3, andTheorem (10.6).

6. In the case of the first problem, for example, the solution involvesfinding two distinct solutions of the system Xl + 2X2 - X3 = - I,2Xl + X2 + 4X3 = 2. This is done by the methods of the previoussections.

7. The line through p and q consists of all vectors p + >.(q - p), by Exercise4. By (10.2), q - p belongs to the directing space of V, and hencep + >.(q - p) E V for all >..

9. Dim (Sl + S2) = 4, dim (Sl n S2) = dim Sl + dim S2 - dim (Sl + S2)

= 2.

10. By Exercise 4, a typical point on the line has the form X = P + >.(q - p),where p = <I, -1,0) and q = <-2, I, I). Substitute the coordinatesof x in the equation of the plane and solve for >..

SECTION 11

1. The mappings in (c), (d), (e) are linear transformations; the others arenot.

3. 2T: Yl = 6Xl + 2X2

Y2 = 2Xl - 2X2

T - V: Yl = -4Xl

Y2 = -X2

T 2:Yl = 10xl - 4X2

Y2 = -4Xl + 2X2

To find the system for TV, we let V: (Xl, X2) --->- (Yl ,Y2) where Yl =

Xl + X2, Y2 = Xl ; T: (Yl , Y2) -> (Zl , Z2) where Zl = - 3Yl + Y2, Z2 =Yl - Y2· Then TV: (Xl, X2) -> <Zl , Z2), where Zl = - 2Xl - 3X2,

Z2 = X2. TV =I- VT.

4. (DM)f(x) = D [ xf(x) ] = xf'(x) + f(x).MDf(x) = (Mf')(x) = xf'(x). DM =I- MD

5. From 0 + 0 = 0, we obtain T(O) + T(O) = T(O), hence T(O) = O.From v + (- v) = 0, we obtain T(v) + T( - v) = T(O) = 0, henceT( - v) = - T(v).

6. The linear transformations defined in (a) and (c) are one to one; theothers are not.

8. The linear transformations defined in (a) and (b) are onto; the others arenot.

9. Suppose T is one-to-one. Then the homogeneous system of equationsdefined by setting all YI'S equal to zero, has only the trivial solution.Therefore the rank of the coefficient matrix is n, and it follows fromSection 8 that T is onto. Conversely, if T is onto, the rank of the coefficient matrix is n, and from Section 9, the homogeneous system hasonly the trivial solution. Therefore T is one-to-one. These remarksprove the equivalence of parts (a), (b) and (c). The equivalence with(d) is proved in Theorem (11.13).


10. D maps the constant polynomials into zero, so that D is not one-to-one./ maps no polynomial onto a constant polynomial # 0, so that / is notonto. The equation DJ = 1 implies that D is onto and that / is one-toone.

-D1

-22

10) (12 1 = 11 3 2

2 3) (;1 1 2

o 0) ( 1-1 0 -1o 2 1

(=~ ~) (~ ~) = (=: -~)

_~) = G(~ ~) (~ ~) = (~ ~)

SECTION 12

1.

4. Multiplying DA is equivalent to multiplying the ith row of A by OJ,while multiplying on the right multiplies the ith column by Of.

5. If A = (au) commutes with all the diagonal matrices D, then by Problem5, we have Oialf = aijof for all OJ and Of in F. If i # j, it follows thatalj = O.

7. The matrices in (b), (c), (d) and (e) are invertible; (a) is not. Check theformulas for the inverses by matrix multiplication.

8. Every linear transformation on the space of column vectors has the formx -+ B . X for some n-by-n matrix B. The linear transformationx -+ A . x is invertible if and only if there exists a matrix B such thatB(A . x) = A(B . x) = x for all x. By the associative law (Exercise 3),these equations are equivalent to (BA)x = (AB)x = x for all x. Thenshow that for an n-by-n matrix C, Cx = x for all x is equivalent toC = I. Thus the equations simply mean that the matrix A is invertible.If A is invertible, then x = A -1b is a solution of the equation Ax = bbecause A(A -1b) = (AA -1)b = Ib = b by the associative law. If x'is another solution, then Ax = Ax', and multiplying by A -1, we obtainA -1(Ax) = A -1(Ax). It follows that x = x'.

SECTION 13

1. The matrices with respect to the basis {U1 , U2} are

S = ( 1 1)-1 0 ' T = (~ ~),

With respect to the new basis {W1 , W2} the matrix of S is

S' = (475 i) = X-1SX, where X = ( 3 1)

-4 -t -1 1 .


2. a b c d

S rank I, nullity I not invertible UI + Uz UI + Uz

T rank 2, nullity 0 invertibleU rank 3, nullity 0 invertible

3.

(;I 0

}0 2

D= rank = k,nullity = I.

4. The rank of T is the dimension of T( V). A basis for T( V) can beselected from among the vectors T(c), i = I, ... , n. Since T(Vi) =Li "i;Wi , a subset of the vectors {T(Vi)} forms a basis for T( V) jf and onlyif the corresponding columns of the matrix A of T form a basis for thecolumn space of A. Therefore dim T( V) = rank (A).

5. Since dim RI = I, the rank of T is one. Therefore n(f) = n - I, byTheorem (13.9).

6. Let VI = n(f); then dim VI = n - I, by Exercise 5. Let (·0 be a fixedsolution of the equation f(v) = ". (Why does one exist?) Then the setof all solutions is Vo + VI, and is a linear manifold of dimensionn - I.

9. If TS = 0 for S -:f. 0, then VI = S(v) -:f. 0 for some V E V, and TVI = O.Conversely, let T(VI) = 0, for some nonzero vector VI. There exists abasis {VI, Vz, ... , Vn} of V starting with VI. Define S on this basis bysetting S(VI) = VI, S(vz) = ... = S(vn) = O. Then S -:f. 0 and TS = O.

10. Let ST = I. Let {VI, ... , vn} be a basis for V. Since ST = I, it followsthat {TvI , ... , Tv n} is also a basis of V. On each of these basis elements,TS(Tv;) = n,;, so that TS agrees with the identity transformation on abasis. Therefore TS = I.

SECTION 14

1. (e) Two vectors are perpendicular if and only if the cosine of the anglebetween them is zero. From the law of cosines, we have

lib - al1 2 = IIal1 2 + lib liz - 211all Ilbll cos 8.

L1-ab

Since

lib - al1 2 = (b - a, b - a) = IIal1 2 + Ilbll z - 2(a, b)

334

we have

SOLUTIONS OF SELECfED EXERCISES

(a, h)cos 8 = Mlibii .

Therefore a ...L h if and only if (a, b) = O.(f) Iia + hll = Iia - bll if and only if (a + h, a + b) = (a - b, a - h).This statement holds if and only if (a, b) = O.

2. (a) If x belongs to both p + Sand q + S, then x = p + Sl = q + S2,

with Sl and S2 E S. It follows that p Eq + Sand q EP + S, and hencethat p + S = q + S.(b) The set L = p + S of all vectors of the form {p + .\(q - p)} is easilyshown to be a line containing p and q. Let L' = p + S' be any linecontaining p and q. Then q - pES' and since Sand S' are onedimensional S = S'. Since the lines Land L' intersect and have thesame one-dimensional subspace, L = L' by (a).(c) p, q, r are collinear if they belong to the same line L = p + S. Inthat case q - pES and q - rES and since S is one-dimensional,S(q - p) = S(q - r). Conversely, if S(q - p) = S(q - r), the linesdetermined by the points p and q, and q and r have the same onedimensional spaces. Hence they coincide by (a).(d) Let L = p + Sand L' = p' + S' be the given lines. From parts(a) and (b) we may assume S ¢ S'. Since dim S = dim S' = 1,S II S' = O. If Xl and X2 belong to L II L', then Xl - X2 E S II S',hence X, = X2. In order to show that L II L' is not empty, let S = S(s),S' = S(s'); then we have to find .\ and A' such that p + As = p' + A's',and this is true since any 3 vectors in R2 are linearly dependent.

3 \ -(p - r, q - p) Th d' I d' .• 1\ = Ilq _ pl12 • e perpen ICU ar Istance IS

Ilu - rll = lip - r - (p - r,q - p) II: =:11211.

SECTION 15

1 11. (a) ,/_<1,1, I,D), ./_(5, -I, -4, -3).

v 3 v 51

2. 1, v12 (x - 1), a(x2 - X + i),where a is chosen to make Ilfll = 1.

3. The distance is given by II v - (v, uI)udl, where v and UI are as follows:(a) v = (-I, -1>,uI = (2, -1>/v5;(b) v = <1,0>, UI = (I, 1>/V2;(c) v = (2,2>, UI = <1, -1>/V2.

5. (a) (v, w) = 21. j t,'YJj(UI, Uj) = 2 tj'T}I'

(b) Let v = 2r= I ttUt. Then (v, Uk) = 2r= I tl(UI' Uk) = t"

SOLUTIONS OF SELECl'ED EXERCISES 335

6. By the Gram-Schmidt theorem, there exists an orthonormal basis{UI , ... , un} of R n starting with UI. The matrix with rows UI, ... , Un

has the required properties.

9. Let {VI, ... , vn} be an orthonormal basis of V, and let

be a basis for W. A vector x = XlVI + ... + XnV n belongs to W.L if andonly jf (x, WI) = ... = (x, Wd) = O. Thus Xl' ... ,Xn have to be solutions of the homogeneous system

The result is now immediate from Corollary (9.4).

10. It can be shown that there exist orthonormal bases {Vj} and {WI} of V suchthat {VI, ... , Vd} is a basis for WI, and {WI, ... , Wd} is a basis for W 2 .

By Theorem 15.11, there exists an orthogonal transformation T suchthat TVt = WI for each i. Then T( WI) = W2 •

11. (a) By Exercise 7, dim S(n).L = 2. It is sufficient to prove that the setP of all p such that (p, n) = a is the set of solutions of a linear equation.Let {VI, V2, V3} be an orthonormal basis for R3 , and let n = alVI +a2V2 + a3V3. Then X = XlVI + X2V2 + X3V3 satisfies (p, n) = a if andonly if alXI + a2X2 + a3X3 = a. This shows that the set of all p suchthat (p, n) = a is a plane. Using the results of Section 10, we can assertthat P = Po + S(n)J., where Po is a fixed solution of (p, n) = a.(b) Normal vector: <3, -I, 1>.(c) Xl - X2 + X3 = - 2, or (n, p) = - 2.(d) The plane with normal vector n, passing through p is the set of allvectors X such that (x, n) = (p, n), or (x - p, n) = O.(f) The normal vector n must be perpendicular to (2, 0, - I) - <t, I, 1>and to (2,0, -I) - (0,0, 1> by (d). Then use the method of part (e).(g) Following the hint, we write the second equation in the formPo - P = >'n + (u - p). Taking the inner product with n yields 0 =

>'(n, n) + (u - p, n).

(h) We have U = <t, 1,2>, p = (0,0,1>, n = <t, I, -1>. Then 3>. +(u - p, n) = 0, >. = -to Then the distance is Ilpo - ull = tllnll.

SECTION 16

1. (a) 2;(b) 0; (c) O;(d) the determinants are: (a) O,(b) 1,(c) 13,(d) -2,(e) 3.


2. Using the properties of the determinant function, we have D(A) =ai ... anD(A'), where

'-(~~ ... *)A - '., .. . . *0···0 I

Then show that, by applying elementary row operations to A', we obtain

D(A') =

o

o *I *

o

**

1 0010

o

oo

3. Show that by applying row operations of type II only (see Definition (6.9»to the rows involving At. we obtain

D(A) =

(XlI • **

0" . *aldI

0 A2

0 0 A,

for some scalars all, .. " aidi (where Ai is a di-by-di matrix) andD(Ad = all ... aidi' Similarly we can apply elementary row operationsto the rows involving A2 , and obtain

(XII • *•

0·· . *aidl

D(A) = 0:21 • *0

o... a;d2

0 0

where a2i ... a2d2 = D(A2 ). Continuing in this way, A is reduced totriangular form. Applying Exercise 2, we obtain the final result.

4. All that has to be proved is that

D*(al , ... , an) = D*(al' ... , at + aj, ... , an)

and this is immediate from assumptions (c) and (d) about D*.

SECTION 17


1. Using the definition and Theorem 16.6, we have D«g, 'rJ), <>., ",,» :.=

D«g, 0), <>., 0» + D«g, 0), <0, ",,» + D«O, 'rJ), <>',0» + D«O, 'rJ),<0, ",,» = g"" - 'rJ>.. D«g, 0), <>., 0» = 0 because the vectors involvedare linearly dependent. D(<~, 0), <0, J.L» = ~J.L D(elo e2), while D(<0, 7]),

<'\,0» = - D«>', 0), <0, 'rJ» = - >''rJ.

SECTION 18

2. AA - 1 = I. Apply Theorem (18.3) to this equation.

3. D(P1j) = - I; D(B jj(>'» = I; D(D j(",,» = "".The fact that A is a product of elementary matrices was shown inSection 12.

5. Let T be an orthogonal transformation. If A is the matrix of T withrespect to an orthonormal basis, then AlA = I, and since D(A) =D(tA) we have D(A)2 = I.

SECTION 19

4. Expanding the determinant along the first row, we see that it has theform AXI + BX2 + C. Substituting (a, f3) or (y, 8) for (Xl, X2) makestwo rows of the determinant equal, and hence the equation AXI + BX2

+ C = °is satisfied by the points (a, /3) and (y, 8).

6. The image of the square is the parallelogram {,\T(el) + ""T(e2): 0 :5 >.,"" :5 I}. The area of the parallelogram is ID(T(el)' T(e2»I, whereT(el) and T(e2) are the columns of the matrix of T with respect to thebasis {el, e2} of R2.

7. Since

al a2/31 /32Yl Y2

the determinant is

al a2 1/31 - a1 /32 - a2 0,Y1 - a1 Y2 - a2 °

1

/31 - al /32 - a2 !,Y1 - al Y2 - a2

which is the area of the parallelogram with edges </3, , /32) - <al' a2>'<Yl' Y2) - <al' a2). The area of the triangle is one-half the area of thisparallelogram.

11. There exists an orthogonal transformation T such that T(adilalll)= e1, ... , T(an/ilanII) = en, where {el'.'" en} are the unit vectors.Then D(al'" .,an) = ID(T)ID(el'" .,en)lIalll ... Ilanll = lIa111 ...Ilanll-

338

SECTION 20

SOLUTIONS OF SELECTED EXERCISES

1. Q = tx - t, R = -tx2 - X - t.2. Let "" ... ,"Ie be distinct zeros of f Then I = (x - ",)g,. Since

"2 "# "" x.- "2 is a prime polynomial which divides Ibut not x - ",.Therefore x - "2 divides g, and I = (x - ",)(x - "2)g2. Continuingin this way, I = (x - ",) ... (x - "Ie)g, and hence deg I ~ k.

4. If the degree oflis two or three, any nontrivial factorization will involvea linear factor, and hence a zero of f The result is false if deg I> 3.For example, I = (x 2 + 1)2 has no zeros in R, but is not a prime inR [x].

5. Suppose min satisfies the equation. Multiplying the resulting equationby n' we obtain

Then n divides aom' , and since n does not divide m', nlao. Similarlymla,.

7. In Q [ x ] , the prime factors are: (a) (2x + 1)(x2 - X + I); (c) (x 2 + I)(x 4 - x 2 + I).In R [x], the prime factors are: (a) (2x + l)(x2 - X + I); (c) (x2 + 1)(x2 + V} x + 1)(x2 - V} x + I).

8. The process must terminate, otherwise we have an infinite decreasingsequence of nonnegative integers, contrary to the principle of wellordering. Now suppose rio "# 0 and rto+' = O. From the way therl's are defined, rto Irio and rio -,. From the preceding equation we seethat rial rio _2. Continuing in this way we obtain rtola and rtolb. On theother hand, starting from the top, if dla and dlb, then dlro. From thenext equation we get dlr,. Continuing we obtain eventually dlrto'Thus rto = (a, b).

9. (a) 2x + I.

SECTION 21

1. -10 + \Oi, YJ(3 - 2i),'W + 4i).

2. cos 38 = (cos 8)3 - 3 cos 8(sin 8)2, obtained by taking the real part ofboth sides in the formula (cos 8 + i sin 8)3 = cos 38 + i sin 38.

3. .\(cos 81e + i sin 81e) where .\ is a real fifth root of 2 and 8k = 21Tk/5,k = 0, 1,2, 3,4.

5. The one-to-one mapping that produces the isomorphism is

(" -P,,)'" + iP --+ P

9. The equation x 2 = - 1 has a solution in the field of complex numbers,but cannot have a solution in an ordered field.

SECTION 22


2. Let {VI, .... ,vm} be a basis for V and {WI, ... , Wn } a basis for W.For each pair (i, j), let Ell be the linear transformation such thatEijvl = WI' and EIIVk = 0 if k ¥- j. Then the mn linear transformations{Ell} form a basis for L( V, W).

3. The minimal polynomials are (x - 2)(x + 1), x 3- 1, x 2 + X - I, and

x 3 , respectively.

4. (a) For each VI, f(T)vl = (T - tl) ... (T - en)vl = 0 since the factorsT - tk commute, and (T - tl)VI = o.(b) Let m(x) = n(x - ~), where the ~j are the distinct eigenvalues of T.By the argument of part (a), m(T) = O. Then the minimal polynomial ofT divides m(x). It is enough to show that if m'(x) = n~j"~k(x - ~), thenm'(T) ¥- O. We have

and the result is proved.

5. Suppose T is invertible, and let m(x} = x r + ... + alX + ao be theminimal polynomial. Suppose ao = O. Then m(x) = xml(X). Moreover, ml(T) ¥- 0 since m(x) is the minimal polynomial. Then m(T) =

Tml(T) = 0, contradicting the fact that T is invertible.Conversely, suppose (to ¥- O. Then m(x) = ml(x)'x + (to for somepolynomial ml(x), and ml(T)T = -(t01. Then -aiiIml(T) = T-I.

6. (a) Suppose a is an eigenvalue of T. Then for some V¥- 0, we haveTv = av. Then a¥-O (why?), and T-ITv = V = aT-Iv. Then T-Iv =a-Iv. The proof of the converse is the same.(b) Suppose Tv = aV. Then f(T)v = f(a)v.

8. This result follows from the fact that similar matrices can be viewed asmatrices of a single linear transformation with respect to different bases,and that their eigenvalues are the eigenvalues of the linear transformation.A direct proof can be made as follows. Let A = S -I BS for some invertiblematrix S. If x¥-O and Ax = ax, then it follows that B(Sx) = a(Sx) andSx ¥- O.

9. The eigenvalues of the given matrices are as follows: (a) 3, - I, (b) - I, - I;and (c) I, -2.The eigenvectors are found by solving the equations

where A is the given matrix, and a an eigenvalue of A. Be sure to checkyour results.

10. Suppose Tv = av, for a ¥- O. Then 0 = Tm = amv, and hence a = O.

11. and 12. are both easy consequences of Theorem (22.8).

340

SECTION 23

SOLUTIONS OF SELECTED EXERCISES

2. The minimal polynomial of G =~) is x2 + 1, which can be factored

into distinct linear factors in C [ x ] but not in R [ x ] .

3. The minimal polynomial is x 3 - 1 = (x - 1)(x2 + X + 1). Thereasoning from this point on is the same as in Exercise 2.

4. The minimal polynomial of D is x n + I .

8. Since d(x)lf(x), n [ d(T) ] c n [J(T) ] . Conversely, let f(T)v = O..Since d(x) = a(x)m(x) + b(x)f(x) for some polynomials a(x) and b(x),we have

d(T)v = a(T)m(T)v + b(T)f(T)v = 0,

since f(T)v = 0 and since m(x) is the minimal polynomial. Thusn [J(T) ] c n [ d(T) ] and the result is established.

SECTION 24

1. (a) (x + 2)2.(b) (x + 2)2.(c) - 2 (appearing twice in the characteristic and minimal polynomials).(d) No. Because the minimal polynomial is not a product of distinctlinear factors.(e) Let v = XlVI + X2V2 be a vector with unknown coefficients suchthat (T + 2)v = O. Using the definition of T, this leads to a system ofhomogeneous equations with the nontrivial solution, (1, -1). ThenVI - V2 is an eigenvector of T.(f) A basis which puts the matrix of Tin triangular form is WI = VI - V2,

W2 = V2. Then S8 = AS where

(-2

8= o

(g) The Jordan decomposition of Tis T = D + N, where D and N are

the linear transformations whose matrices are - 21 and (~ ~) with

respect to the basis {WI, W2}.

3. The minimal polynomial of T is x 2 + afJ, where afJ > O. The minimalpolynomial is not a product of distinct linear factors in R [ x ] , and theanswer to the question is no.

4. Use the triangular form theorem.5. The minimal polynomial of T divides x 2 - x, and therefore has distinct

linear factors in C [ x ]. There does exist a basis of V consisting of eigenvectors.


6. The minimal polynomial divides x' - 1 which has distinct linear factorsin C [x] , namely x - (cos Ok + i sin Ok), k = 0, ... , r - 1, where Ok =

21Tk/r.

7. V has a basis consisting of eigenvectors of T if and only if A. oF O.

SECTION 25

)-1 0 0 0 0

0 0 -1 0 10 1 -1 0 0

0 0 0 -11 -1

(b)

1. (a)

2. The rational canonical forms are

(~ ~) , (i0

-~) ,01 -1

respectively.

3. The rational canonical forms over Rare

(~+ -; t _0-;) G0

-~) ,and 01 -1

respectively.

4. The canonical forms over Care

ando 0 )-t + ../-3 0

o 2 -t _ ../;3

respectively.

5. Let {v, Tv, ... , T d-IV} be a basis for V, and suppose TdV = aoV +alTv + ... + ad_lTd-Iv. The matrix of T with respect to this basis is

A = (~ ..~ .. : : : .. ~~ .. ) .

o ... ad-I

Then D(A - xl) = ± (xd - ad _ IXd - I - .•• - ao). On the other hand,x d - ad _ IXd - I - ... - ao is the minimal polynomial of T by Lemma(25.6).

6. The matrices are similar over C in all three cases.

Symbols(Including Greek Letters)

Lower-case Greek Letters Used in This Book

IX alphap beta

'Y gammacS delta€ epsilon, zeta

7J eta(J theta,\ lambda

fI- muv nu~ xi (ksi)71' pip rhoG sigmaT tauq> phi

ifJ psiw omega

342

Partial List of Symbols Used

SYMBOLS 343

RaEAAcBf: A-+B

<<Xl'···' <xn>L: XI

TI Ui

S(V I , ... , Vn)§,(R)C(R)P(R)L(V, W)A, aA-BIA,IT

l<xl(u, v)

IlullD(u l , ... , un), D(A),

det A, D(T)Tr (A), Tr (T)neT), T(V)

F [x]Z

VI EEl V2

f(T)Sl.

eA

V*V X WV® W, T® U,

AxB/\kY,V/\WT'

field of real numbersset membershipset inclusionfunction (or mapping) from A into Bvector with components {<Xl' ... , <xn}.

summationproductvector space generated (or spanned) by {VI' ... , Vn}.vector space of all real-valued functions on R.continuous real valued functions on Rpolynomial functions on Rlinear transformations from V into Wmatrices (write by hand ~, l!)row equivalence of matricestranspose of a matrix A, of a linear transforma-

tion Tabsolute valueinner productlength of a vector U

determinant of a set of vectors, of a matrix, of alinear transformation

trace of a matrix A, or of a linear transformation T.null space, range of a linear transformation

T: V-+ W.polynomials with coefficients in a field Fconjugate of a complex number zdirect sum of vector spacespolynomial in a linear transformation Tset of vectors orthogonal to the vectors in Sexponential of a matrix Adual space of Vcartesian product of Vand Wtensor product of vector spaces, linear transforma

tions, matrices.wedge product of vector spaces, vectors.adjoint of a linear transformation

Index

A

Absolute value, 178Algebraically closed field, 180Angle, 123Associative law for matrix multiplication,

92Augmented matrix, 56

B

Basis of vector space, 36Bilinear form, 236

nondegenerate, 237skew symmetric, 243symmetric, 243vector spaces dual to, 239

Bilinear function, 119,245Binomial coefficient, 15

c

Cartesian product, 243Cauchy-Schwartz inequality, 120Cayley-Hamilton theorem, 206Characteristic polynomial, 205Characteristic root, 189Characteristic vector, 189Coefficient matrix, 53Cofactor, 150Column expansion, 150Column subspace, 54Column vector, 54Commutative ring, 81Companion matrix, 221, 222

344

Complex number, 177conjugate of, 178DeMoivre's theorem, 180roots of unity, 180

Composition of sums of squares, 317Conjugate, 178Cramer's rule, 152Cyclic group en, 114Cyclic subspace, 217

D

DeMoivre's theorem, 180Determinant, column expansion of, 150

as volume function, 154complete expansion of, 144, 159definition of, 134Hadamard's inequality, 155minor, 153row expansion of, 151van der Monde, 161

Diagonalizable linear transformation, 198Diagonal matrix, 198Dihedral group Dn, 114Dimension of vector space, 37Direct sum, 195,243Division process (for polynomials), 167Dual space, 234

and bilinear form, 239

E

Echelon form, 44Eigenvalue, 189Eigenvector, 189

INDEX

Elementary divisor theorem, 219Elementary divisors, 220, 223Elementary matrix, 94Elementary row operation, 43Equivalence relation, 228Exponential matrix, 300

F

Factor theorem, 169Field, algebraically closed, 180

definition of, 8of rational functions, 174ordered, 9quotient, 174

Finite groups of rotations, 292Function, bilinear, 119, 245

linear, 234one-to-one, 76onto, 76polynomial, 27positive definite, 119vector space, 19

G

Gaussian elimination, 40Generators of subspace, 28Gram-Schmidt orthogonalization

process, 124Graph, 212Greatest common divisor, 170Group, cyclic, 114

definition of, 82dihedral, 114finite rotation, 292symmetry, 112

H

Hadamard's inequality, 155Hermitian linear transformation, 283Hermitian scalar product, 279Homogeneous equations, 54

Idempotent linear transformation, 196Identity matrix, 93Incidence matrix, 212Indecomposable subspace, 261Inequality, Cauchy-Schwarz, 120

Hadamard, 155triangle, 121

Inner product, 119

Invariant subspace, 194Inverse linear transformation, 81Invertible linear transformation, 81Invertible matrix, 94Irreducible polynomial, 170Irreducible vector space, 265Isometric vector space, 130Isomorphism, of fields, 10

of vector spaces, 84

J

Jordan canonical form, 224Jordan decomposition, 206Jordan normal form, 224

K

k-fold tensor product, 255k-fold tensors, 255Kronecker product, 250

L

Linear dependence, relation of, 30Linear function, 234Linear independence, 30Linear transformation, characteristicpolynomial of, 205

definition of, 76diagonalizable, 198hermitian, 283idempotent, 196inverse of, 81invertible, 81minimal polynomial of, 186nilpotent, 203nonsingular, 81normal, 283null space of, 106product of, 78reflection of, 116,290rotation of, 116, 270self-adjoint, 283sum of, 78symmetric, 274trace of, 210transpose of, 234unitary, 280

Linearly dependent vectors, 30Linearly independent vectors, 30

M

Markov chain, 3I2Matrix, 38

augmented, 56

345

346

characteristic polynomial of, 205coefficient, 53column subspace of, 54column vectors of, 54companion, 221,222definition of, 39diagonal,198elementary, 94exponential, 300identity, 93incidence, 212invertible, 94Jordan normal form of, 224m-by-n, 39multiplication of, 92orthogonal, 129product, 90rank of, 56row subspace of, 54row vectors of, 53similarity of, 104stochastic, 312transpose of, 146

m-by-n matrix, 39Minor determinant, 153Multiplication of matrices, 92Multiplication theorem, 148

N

Nilpotent linear transformation, 203Nondegenerate bilinear form, 237Nonhomogeneous system, 54

Cramer's rule, 152Nonsingular linear transformation, 81Nontrivial solution, 54Normal linear transformation, 283Normal vector, 130n-tuple, 18Nullity, null space, 106

o

Order, definition of, 218Ordered field, 9Orthogonal matrix, 129Orthogonal transformation, 127Orthonormal basis, 123Orthonormal set, 123

P

Permutation, 156signature of, 158

Perron-Frobenius Theorem, 310Polar decomposition, 287

INDEX

Polynomial, characteristic, 205definition of, 163degree of, 166irreducible, 170minimal, 186

Polynomial equation, 169Polynomial function, 27, 168Positive definite function, 119Primary decomposition theorem, 196Prime, 170

relative, 170Principal axes, 273Principle of mathematical induction, 10Product matrix, 90Proper value, 189Proper vector, 189

Q

Quadratic form, 271Quotient field, 174Quotient set, 230Quotient space, 231

R

Range, 106Rank,56

column, 64of matrix, 56row, 64

Rational canonical form, 220Reflection, 116,290Relatively prime elements, 170Remainder theorem, 169Restriction, 231Ring, 80

commutative, 81Root, characteristic, 189

of unity, 180Rotation, 116, 270

finite groups of, 292Row equivalence, 43Row expansions, 151Row vectors, 53Row subspace, 54

sSelf-adjoint linear transformation, 283Set, orthonormal, 123Signature, 158Similar matrices, 104Skew symmetric bilinear form, 243Skew symmetric tensors, 256

INDEX

Solution, nontrivial, 54trivial, 54

Solution space, 62Solution vector, 54Spectral theorem, 285Spectrum, 284, 306Subfield,9SUbspace, 26

column, 54cyclic,217dual,234finitely generated, 28generators of, 28indecomposable, 261invariant, 194quotient, 231row, 54spanned by vectors, 28

Sylvester's Inertia Theorem, 276Symmetric bilinear form, 243Symmetric transformation, 274Symmetry group of figure, 112System of linear equations, coefficientmatrix of, 53Cramer's rule, 54first-order differential, 298homogeneous, 54min n unknowns, 53nonhomogeneous, 54

T

Tensor, k-fold, 255skew symmetric, 256

Tensor product, 249,250k-fo1d,255

Trace of linear transformation, 210Transpose, of linear transformation, 234

of matrix, 146Transposition, 158Triangle inequality, 121

Triangular form theorem, 202Trivial solution, 54

u

Unique factorization theorem, 171Unit, 169Unit vectors, 37Unitary transformation, 280

v

van der Monde determinant, 161Vector, eigen-, characteristic, 189

column, 54echelon form, 44length,120linearly dependent, 30linearly independent, 30orthonormal set, 123proper, 189row,53solution, 54unit, 37

Vector space, 18basis of, 36completely reducible, 265dimension of, 37dual,239Fw 19of functions, 19iIieducible, 265isometric, 130R n,19

Volume function, 154

wWedge product, 256WeU-ordering principle, 10

347

Undergraduate Texts in Mathematics

(continued from page ii)

Gamelin: Complex Analysis.Gordon: Discrete Probability.HairerlWanner: Analysis by Its History.

Readings in Mathematics.Halmos: Finite-Dimensional Vector

Spaces. Second edition.Halmos: Naive Set Theory.Hlimmerlin/Hoffmann: Numerical

Mathematics.Readings in Mathematics.

Harris/HirstiMossinghoff:Combinatorics and Graph Theory.

Hartshorne: Geometry: Euclid andBeyond.

Hijab: Introduction to Calculus andClassical Analysis.

Hilton/Holton/Pedersen: MathematicalReflections: In a Room with ManyMirrors.

Hilton/Holton/Pedersen: MathematicalVistas: From a Room with ManyWindows.

Iooss/Joseph: Elementary Stabilityand Bifurcation Theory. Secondedition.

Isaac: The Pleasures of Probability.Readings in Mathematics.

James: Topological and UniformSpaces.

Jlinich: Linear Algebra.Jlinich: Topology.Jlinich: Vector Analysis.Kemeny/Snell: Finite Markov Chains.Kinsey: Topology of Surfaces.KIambauer: Aspects of Calculus.Lang: A First Course in Calculus. Fifth

edition.Lang: Calculus of Several Variables.

Third edition.Lang: Introduction to Linear Algebra.

Second edition.Lang: Linear Algebra. Third edition.Lang: Short Calculus: The Original

Edition of "A First Course inCalculus."

Lang: Undergraduate Algebra. Secondedition.

Lang: Undergraduate Analysis.Laubenbacher/Pengelley: Mathematical

Expeditions.Lax/BursteinlLax: Calculus with

Applications and Computing.Volume 1.

LeCuyer: College Mathematics withAPL.

LidllPilz: Applied Abstract Algebra.Second edition.

Logan: Applied Partial DifferentialEquations.

LovaszlPelikanNesztergombi: DiscreteMathematics.

Macki-Strauss: Introduction to OptimalControl Theory.

Malitz: Introduction to MathematicalLogic.

MarsdenIWeinstein: Calculus I, II, III.Second edition.

Martin: Counting: The Art ofEnumerative Combinatorics.

Martin: The Foundations of Geometryand the Non-Euclidean Plane.

Martin: Geometric Constructions.Martin: Transformation Geometry: An

Introduction to Symmetry.Millman/Parker: Geometry: A Metric

Approach with Models. Secondedition.

Moschovakis: Notes on Set Theory.Owen: A First Course in the

Mathematical Foundations ofThermodynamics.

Palka: An Introduction to ComplexFunction Theory.

Pedrick: A First Course in Analysis.PeressinilSullivanlUhl: The Mathematics

of Nonlinear Programming.PrenowitzlJantosciak: Join Geometries.Priestley: Calculus: A Liberal Art.

Second edition.

Undergraduate Texts in Mathematics

Protter/Morrey: A First Course in RealAnalysis. Second edition.

Protter/Morrey: Intermediate Calculus.Second edition.

Pugh: Real Mathematical Analysis.Roman: An Introduction to Coding and

Information Theory.Ross: Elementary Analysis: The Theory

of Calculus.Samuel: Projective Geometry.

Readings in Mathematics.Saxe: Beginning Functional AnalysisScharlau/Opolka: From Fermat to

Minkowski.Schiff: The Laplace Transform: Theory

and Applications.Sethuraman: Rings, Fields, and Vector

Spaces: An Approach to GeometricConstructability.

Sigler: Algebra.SilvermanlTate: Rational Points on

Elliptic Curves.Simmonds: A Brief on Tensor Analysis.

Second edition.Singer: Geometry: Plane and Fancy.Singer/Thorpe: Lecture Notes on

Elementary Topology andGeometry.

Smith: Linear Algebra. Third edition.Smith: Primer of Modem Analysis.

Second edition.StantonlWhite: Constructive

Combinatorics.Stillwell: Elements of Algebra: Geometry,

Numbers, Equations.Stillwell: Elements of Number Theory.Stillwell: Mathematics and Its History.

Second edition.Stillwell: Numbers and Geometry.

Readings in Mathematics.Strayer: Linear Programming and Its

Applications.Toth: Glimpses of Algebra and Geometry.

Second Edition.Readings in Mathematics.

Troutman: Variational Calculus andOptimal Control. Second edition.

Valenza: Linear Algebra: An Introductionto Abstract Mathematics.

WhyburnlDuda: Dynamic Topology.Wilson: Much Ado About Calculus.

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