Vector Spaces
Section 3.5 Computing with Bases 93
3.6 DIRECT SUMS
The concepts of independence and span of a set o f vectors have analogues for subspaces.
If W i, • . . , Wk are subspaces of a vector space V, the set of vectors v that can be written as a sum
(3.6.1) v = w i +--- + Wh
where w, is in Wi is called the sum of the subspaces or their span, and is denoted by Wi + . • • + Wk:
(3.6.2) Wi +---- + Wk = {v e V | v = w i + ---+ Wh with w; in W(}.
The sum of the subspaces is the smallest subspace that contains all of the subspaces Wi . . . . , Wk. It is analogous to the span of a set of vectors.
The subspaces Wi, . . . , Wk are called independent if no sum wi + ---- + Wk with w; in Wi is zero, except for the trivial sum, in which w; = 0 for all i. In other words, the spaces are independent if
(3.6.3) w i + ---- + Wk = 0, with w; in Wi, implies w; = 0 for all i.
Note: Suppose that vi, . . . , Vk are elements of V, and let W, be the span of the vector u,. Then the subspaces Wi, . . . , Wk are independent if and only if the set (v i, . . . , v„) is independent. This becomes clear if we compare (3.4.8) and (3.6.3). The statement in terms of subspaces is actually the neater one, because scalar coefficients don’t need to be put in front of the vectors w; in (3.6.3). Since each of the subspaces Wi is closed under scalar multiplication, a scalar multiple cw; is simply another element of W,. □
We omit the proof of the next proposition.
Proposition 3.6.4 Let Wi , . . . , Wk be subspaces of a finite-dimensional vector space V, and let B, be a basis of Wi.
(a) The following conditions are equivalent:
• The subspaces W; are independent, and the sum Wi +---- + Wk is equal to V.
• The set B = (B i, . . . , Bk) obtained by appending the bases Bi is a basis of V.
(b) dim(Wi + ■ ■ ■ + Wk) :: dim Wi + ---+ dim Wk, with equality if and only if the spaces are independent.
(c) If Wi is a subspace of W, for i = 1, ... , k, and if the spaces Wi, . . . , Wk are independent,
then so are the W i, . . . , W^. □
If the conditions of Proposition 3.6.4(a) are satisfied, we say that V is the direct sum of W \ , . . . , Wk, and we write V = Wi EB ■ ■ ■ EB W k
( 3 6 5) V = Wi EB ... EB Wk, if Wi + • • . + Wk = V ( ' ' ' and Wi, . . . , Wk are independent.
If V is the direct sum, every vector v in V can be written in the form (3.6.1) in exactly one way.
Proposition 3.6.6 Let Wi and W2 be subspaces of a finite-dimensional vector space V.
(a) dim Wi + dim W2 = dim (W inW 2) + dim(Wi + W2).
(b) Wi and W2 are independent if and only if Wi 0 W2 = {OJ.
(c) V is the direct sum Wi EB W2 if and only if Wi n W2 = {OJ and Wi + W2 = V.
(d) If Wi + W2 = V, there is a subspace W^ of W2 such that Wi EB W'2 = V.
Proof. We prove the key part (a): We choose a basis, U = (u i, . . . , uk) for W \ 0 W2, and we extend it to a basis (U, V) = (u i, . . . , uk; Vi, . . . , vm) of Wi. We also extend U to a basis (U, W) = (u i, . . " , uk; w i, . . . , w n) of W2. Then dim(W i 0 W2) = k, dim Wi = k + m, and dim W2 = k + n. The assertion will follow if we prove that the set o fk + m + n elements (U, V, W) = (u i, . . . , uk; vi, . . . , Vm; W i ,. . . , wn) is a basis of Wi + W2.
We must show that (U, V, W) is independent and spans Wi + W2. An element v of Wi + W2 has the form w ' + w" where w ' is in W and w" is in W2. We write w ' in terms of our basis (U, V) for Wi, say w ' = UX + VY = uiXi + . • • + ukXk + V1Y1 +---- + VmYm. We also write w" as a combination UX' + WZ of our basis (U, W) for W2. Then V = w' + w" = U (X + X') + VY + WZ.
Next, suppose we are given a linear relation UX + VY + WZ = 0, among the elements (U, V, W). We write this as UX + VY = -WZ. The left side of this equation is in Wi and the right side is in W2. Therefore -W Z is in Wi 0 W2, and so it is a linear combination U X ' of the basis U. This gives us an equation UX' + WZ = O. Since the set (U, W) is a basis for W2, it is independent, and therefore X ' and Z are zero. The given relation reduces to UX + VY = O.
But (U, V) is also an independent set. So X and Y are zero. The relation was trivial. □ 3.7 INFINITE-DIMENSIONAL SPACES
Vector spaces that are too big to be spanned by any finite set of vectors are called infinite
dimensional. We won’t need them very often, but they are important in analysis, so we discuss them briefly here.
One of the simplest examples of an infinite-dimensional space is the space jRoo of infinite real row vectors
(3.7.1) (a) = ( a i , a 2, a 3, . . . ) .
A n infinite vector can be thought of as a sequence a^, «2, of real numbers.
Section 3.7 Infinite-Dimensional Spaces 97 The space jRoo has many infinite-dimensional subspaces. Here are a few; you will be able to make up some more:
Examples 3.7.2
(a) Convergent sequences: C = {(a) e jRoo | the limit lim exists }.
OQ
(b) Absolutely convergent series: £* = {(a) e JROO | L ian I < oo}.
I (c) Sequences with finitely many terms different from zero.
Z = {(a) € RC)O ! = 0 for all but finitely many n}.
Now suppose that V is a vector space, infinite-dimensional or not. What do we mean by the span of an infinite set S of vectors? It isn’t always possible to assign a value to an infinite combination c vi + C2 V2 + .. ' . If V is the vector space ]Rn, then a value can be assigned provided that the series Ci vi + C2V2 + • . converges. But many series don’t converge, and then we don’t know what value to assign. I n algebra it is customary to speak only of combinations of finitely many vectors. The span of an infinite set S is defined to be the set of the vectors v that are combinations of finitely many elements of S:
(3.7.3) v = civ i + ■. + CrW, where u i , . . " , v r are in S.
The vectors v; in S can be arbitrary, and the number r is allowed to depend on the vector v and to be arbitrarily large:
(3.7.4) s pan S = finite combinations
of elements of S
For example, let e = (0, . . . , 0, 1, 0, . . . ) be the row vector in with 1 in the ith position as its only nonzero coordinate. Let E = (ei, e2, e3, . . . ) be the set of these vectors.
This set does not span ]Roo, because the vector
w = (1, 1, 1, . . . )
is not a (finite) combination. The span of the set E is the subspace Z (3.7.2)(c).
A set S, finite or infinite, is independent if there is no finite linear relation (3.7.5) Cjui + • ■' + c^vr = O, with t i , . . . , tv in S,
except for the trivial relation in which Ci = ... = Cr = O. Again, the number r is allowe d to be arbitra ry, that is, the condition has to hold for arbitrarily large r and arbitrary elements vi , . . . , vr of S. For example, the set S' = (w; e i , e2, e3, .. .) is independent, if w and e; are the vectors defined above. With this definition of indep endence, Proposition 3.4.15 continues to be true.
As with finite sets, a basis S of V is an independent set that spans V. The set S = ( e j , e2, . . . ) is a basis of the space Z. The monomials x l form a basis for the space
of polynomials. It can be shown, using Zorn’s Lemma or the Axiom o f Choice, that every vector space V has a basis (see the appendix, Proposition A.3.3). However, a basis for Roo will have uncountably many elements, and cannot be made very explicit.
Let us go back for a moment to the case that our vector space V is finite-dimensional (3.4.16), and ask if there can be an infinite basis. We saw in (3.4.21) that any two finite bases have the same number of elements. We complete the picture now, by showing that every basis is finite. This follows from the next lemma.
Lemma 3.7.6 Let V be a finite-dimensional vector space, and let S be any set that spans V.
Then S contains a finite subset that spans V.
Proof. By hypothesis, there is a finite set, say (wi, . . . , u m), that spans V. Because S spans V, each of the vectors Uj is a linear combination of finitely many elements of S. The elements of S that we use to write all of these vectors as linear combinations make up a finite subset S' of S. Then the vectors w; are in SpanS', and since (wi, . . . , u m) spans V, so does S'. □ Corollary 3.7.7 Let V be a finite-dimensional vector space.
• Every basis is finite.
• Every set S that spans V contains a basis.
• Every independent set L is finite, and can be extended to a basis. □ I don't need to learn 8 + 7: I'll remember 8 + 8 and subtract 7.
—T. Cuyler Young, Jr.
EXERCISES Section 1 Fields
1.1. Prove that the numbers of the form a + b. . , where a and b are rational numbers, form a subfield of C.
1.2. Find the inverse of 5 modulo p, for p = 7, 11, 13, and 17.
1.3. Compute the product polynomial (x3 + 3x2 + 3x + 1) (x4 + 4x3 + 6x2 + 4x + 1) when the coefficients are regarded as elements of the field IF 7. Explain your answer.
1.4. Consider the system of linear equations ' 6 - 3 ' x x V _ 6 3 _ .¾ . 1 (a) Solve the system in Fp when p = 5,11, and 17.
(b) Determine the number of solutions when p = 7.
1.5. Determine the primes p such that the matrix A
is invertible, when its entries are considered to be in Fp.
1 2 0
0 3 -1
-2 0 2
Exercises 99
1.6. Solve completely the systems of linear equations AX = 0 and AX = B, where
'1 1 0 " ‘ 1 '
1 0 1 , and B = -1
1 -1 -1 1
(a) in Q , (b) in lF2 , (c) in F3, (d) inlF7.
1.7. By finding primitive elements, verify that the multiplicative group IF; is cyclic for all primes p < 20.
1.8. Let p be a prime integer.
(a) Prove Fermat’s Theorem: For every integer a, aP = a modulo p.
(b) Prove Wilson’s Theorem, (p — 1)! == -1(modulo p ).
1.9. Determine the orders of the matrices in the group G L2 (IF 7).
1.10. Interpreting matrix entries in th e field lF2, prove that the four matrices
'0 0
'0
)1 Tl 01
0 )J ’ [0 1 _0 ' '1 r ‘0 r
1 ’ 1 0 ’ _1 1 _form a field.
Hint, You can cut the work down by using the fact that various laws are known to hold for addition and multiplication of matrices.
1.11. Prove that the set of symbols {a + bi | a, b e F3} forms a field with nine elements, if the laws of composition are made to mimic addition and multiplication of complex numbers.
Will the same method work for F5? For F 7? Explain.
Section 2 Vector Spaces
2.1. (a) Prove that the scalar product of a vector with the zero element of the field F is the zero vector.
(b) Prove that if w is an element of a subspace W, then -w is in W too.
2.2. Which of the following subsets is a subspace of the vector space F nXn of n x n matrices with coefficients in F?
(a) symmetric matrices (A = A1), (b) invertible matrices, (c) upper triangular matrices.
Section 3 Bases and Dimension
3.1. Find a basis for the space of n X n symmetric matrices (Ar = A).
3.2. Let W C 1R4 be the space of solutions of the system of linear equations A X = 0, where
A 1 2 3
1 2 3 . Find a basis for W.
3.3. Prove that the three functions x2, cos x, and e* are linearly independent.
3.4. Let A be an m X n matrix, and let A' be the result of a sequence of elementary row operations on A. Prove that the rows of A span the same space as the rows of A'.
3.5. Let V = F n be the space of column vectors. Prove that every subspace W of V is the space of solutions of some system of homogeneous linear equations AX = 0.
3.6. Find a basis of the space of solutions in jRn of the equation Xi + 2x2 + 3X3 +--- + nXn = O.
3.7. Let (X i, . . . , Xm) and (Yi, . . . , Yn) be bases for jR™ and jRn, respectively. Do the mn matrices Xi Yj form a basis for the vector space jRm><n of all m X n matrices?
3.8. Prove that a set ( vi , . . . , vn) of vectors in F n is a basis if and only if the matrix obtained by assembling the coordinate vectors of v, is invertible.
Section 4 Computing with Bases
4.1. (a) Prove that the set B = ((1, 2, 0)\ (2, 1, 2 ) \ (3, 1 ,1)‘) is a basis of R3.
(b) Find the coordinate vector of the vector V = (1 ,2 ,3)1 with respect to this basis.
(c) Let B' = ((0; 1, 0)1, (1, 0 , 1)1, (2, 1, 0)1). Determine the basechange matrix P from B to B'.
4.2. (a) Determine the basechange matrix in jRz, when the old basis is the standard basis E = (ei, ez) and the new basis is B = (ei + ez, ei — eZ).
(b) Determine the basechange matrix in jRn, when the old basis is the standard basis E and the new basis is B = {fin, £ n -i, • .. , ei).
(c) Let B be the basis of ]R2 in which vi = ei and Vz is a vector of unit length making an angle of 120° with Vi. Determine the basechange matrix that relates E to B.
4.3. Let B = (vi, . . . , Vn) be a basis of a vector space V. Prove that one can get from B to any other basis B' by a finite sequence of steps of the following types:
(i) Replace v, by v, + avj, i =1= j, for some a in F, (ii) Replace V, by cVj for some c=I= 0,
(iii) Interchange v(- and Vj.
4.4. Let Fp be a prime field, and let V = F2. Prove:
(a) The number of bases of V is equal to the order of the general linear group GLz(lF'p).
(b) The order of the general linear group G Lz(lF'p) is p ( p + l ) ( p — 1)2, and the order of the special linear group SLz(Fp) is p (p + l) ( p — 1).
4.5. How many subspaces ofeach dimension are there in (a) IF'P’ (b) F£?
Section 5 Direct Sums
5.1. Prove that the space ]RnXn of all n X n real matrices is the direct sum of the space of symmetric matrices (A‘ = A) and the space of skew-symmetric matrices (A‘ = -A).
5.2. The trace of a square matrix is the sum ofits diagonal entries. Let Wi be the space of n X n matrices whose trace is zero. Find a subspace W2 so th at jRnXn = Wi ffi W2.
5.3. Let Wi, . . . , Wk be subspaces of a vector space V, such that V = L W Assume that Wi n W2 = 0, ( Wi + W2) n W3 = 0, . . . , ( Wi + W2 +---+ Wk_i) n Wk = 0. Prove that V is the direct sum of the subspaces Wi , . . . , Wk.
Exercises 101
6.1. Let E be the set of vectors (ei, e2, . . . ) in and let w = (1,1,1 , . . . ). Describe the span of the set (w, ei, e2, . . . ) .
6.2. The doubly infinite row vectors (a) = (• • . , a - i, ao, a i, . . . ) , with a,- real form a vector space. Prove that this space is isomorphic to ]Roo.
*6.3. For every positive integer, we can define the space f p to be the space of sequences such that L l a |P < °°. Prove that fP is a proper subspace of £P+1.
*6.4. Let V be a vector space that is spanned by a countably infinite set. Prove that every independent subset of V is finite or countably infinite.
Miscellaneous Problems
M.I. Consider the determinant function d e t: F 2x2 -4 F, where F = IF'p is the prime field of order p and F lx2 is the space of 2 X 2 matrices. Show that this map is surjective, that all nonzero values of the determinant are taken on the same number of times, but that there are more matrices with determinant 0 than with determinant 1.
M.2. Let A be a real n X n matrix. Prove that there is an integer N such that A satisfies a nontrivial polynomial relation A'v + c n -1 An - 1 +--- + Ci A + C o = O.
M.3. (polynomial paths) (a) Let x(t) and y(t) be quadratic polynomials with real coefficients.
Prove that the image of the path (x(t) , y (t)) is contained in a conic, i.e., that there is a real quadratic polynomial f ( x , y) such that f(x ( t) , y(t)) is identically zero.
(b) Let x(t) = t2 — 1 and y(t) = t3 — t. Find a nonzero real polynomial f(x , y) such that f(x (t), y(t)) is identically zero. Sketch the locus {/(x, y) = 0} and the path (x(t) , y ( t»
in ]R2.
(c) Prove that every pair x(t), y(t) of real polynomials satisfies some real polynomial relation f(x , y) = 0.
*M.4. Let V be a vector space over an infinite field F. Prove that V is not the union of finitely many proper subspaces.
*M.S. Let a be the real cube root of 2.
(a) Prove that (1, a, a 2) is an independent set over Q, i.e., that there is no relation of the form a + ba + c a 2 = 0 with integers a, b, c.
Hint: Divide x3 - 2 by cx2 + bx + a.
(b) Prove that the real numbers a + ba + c a 2 with a, b, c in Q form a field.
M.6. (Tabasco sauce: a mathematical diversion) My cousin Phil collects hot sauce. He has about a hundred different bottles on the shelf, and many of them, Tabasco for instance, have only three ingredients other than water: chilis, vinegar, and salt. What is the smallest number of bottles of hot sauce that Phil would need to keep on hand so that he could obtain any recipe that uses only these three ingredients by mixing the ones he had?