If you find a mistake, omission, etc., please let me know by e-mail.

The orange balls mark our current location in the course, and the current problem set.

[No, you don’t have to know French to take Math 55a. Googling

Update 9/9: We now have two CA’s for the class: Wyatt Mackey (

[if writing from outside the Harvard network, append

! If you are coming to class but not officially registered for Math 55 (e.g. you are auditing, or still undecided between 25a and 55a but officially signed up for 25a), send me your e-mail address so that I and the CA's can include you in class announcements.

The Sep.13 office hours will be an hour later than usual, that is,

CA office hours are

Wyatt will hold section

!

Also,

!

!

At least in the beginning of the

- [see Axler, page 5]
*Pace*the boxed note on that page, virtually*all*mathematicians say and write“ (more fully,*n*-tuple”“ordered ), while I cannot recall another instance of “list” used for this as Axler does. (One sometimes sees “tuple” for an*n*-tuple” of unspecified length*n*-tuple*n*, and “ordered pair” and perhaps “ordered triple”, “ordered quadruple”, etc. for*n*= 2, 3, 4, …) - [cf. Axler, Notation 1.6 on page 4, and the
“Digression on Fields” on page 10]

Unless noted otherwise,**F**may be an arbitrary field, not only**R**or**C**. The most important fields other than those of real and complex numbers are the field**Q**of rational numbers, and the finite fields**Z**/*p***Z**(*p*prime). Other examples are: the field**Q**(*i*) of complex numbers with rational real and imaginary parts; more generally,**Q**(*d*^{1/2}) for any nonsquare rational number*d*; the “*p*-adic numbers”**Q**_{p}(*p*prime), of which we’ll say more when we study topology next term; and more exotic finite fields such as the 9-element field (**Z**/*3***Z**)(*i*). Here’s a review of the axioms for fields, vector spaces, and related mathematical structures. - [cf. Axler, p.28 ff.] We define the
*span*of an arbitrary subset*S*of (or tuple in) a vector space*V*as follows: it is the set of all (finite) linear combinations with each*a*_{1}*v*_{1}+ … +*a*_{n}v_{n}*v*in_{i}*S*and each*a*in_{i}*F*. This is still the smallest vector subspace of*V*containing*S*. In particular, if*S*is empty, its span is by definition {0}. We do*not*require that*S*be finite. - Warning: in general the space
*F*[*X*](a.k.a. of polynomials in*P*(*F*))*X*, and its subspaces of polynomials of degree at most*P*_{n}(*F*)*n*, might not be naturally identified with a subspace of the space*F*of functions from^{F}*F*to itself. The problem is that two different polynomials may yield the same function. For example if*F*is the field of 2 elements the polynomial gives rise to the zero function. In general different polynomials can represent the same function if and only if*X*^{2}−*X**F*is finite — do you see why? - (See also Exercise 11 in Axler 1.C, assigned as part of the first
problem set)

If*U*are any subspaces of a vector space_{i}*V*then so is their intersection∩ . Note that this doesn’t specify a finite intersection:_{i}*U*_{i}*i*could range over an “index set”*I*of any cardinality (so we would write the intersection as∩ We don’t usually want to intersect an_{i∈I}*U*)._{i}*empty*family of sets (do you see why?), but for subsets of a given set*V*we can declare that∩ _{i∈∅}*U*=_{i}*V*. - For any field (or even any ring)
*F*there is a canonical ring homomorphism, call it*h*, from**Z**to*F*. “Ring homomorphism” means: ,*h*(0) = 0 , and for any integers*h*(1) = 1 we have*m*,*n* and*h*(*m*+*n*) =*h*(*m*) +*h*(*n*) (and*h*(*mn*) =*h*(*m*)*h*(*n*) , though this follows from the other properties). But this doesn’t quite mean that we get an isomorphic copy of*h*(−*n*) = −*h*(*n*)**Z**in*F*, because*h*might not be injective. Equivalently, the*kernel*(that is, the preimage might be larger than just {0}. In general,*h*^{−1}({0}) = {*n*:*h*(*n*) = 0})*I*must be an*ideal*, i.e. an additive subgroup of**Z**that is closed under multiplication by*arbitrary*integers (whether in*I*or not — this mimics the definition of a subspace, though as it happens for ideals in**Z**it’s automatic). Now ideals in**Z**are either {0} or( for some*n*) := {*cn*|*c*in**Z**}*n*(namely the least positive element of the ideal). For rings, any*n*may arise, most easily for the ring of integers mod**Z**/*n***Z***n*. But if*F*is a field and*I*=(*n*) then*n*must be either zero or prime, lest*F*have zero divisors (elements*a*and*b*, neither zero, for which*ab*=0). This*n*is then called the*characteristic*of the field*F*. The familiar fields all have characteristic zero. For any prime**Q**,**R**,**C***p*, there are fields of characteristic*p*, notably the “prime field” (mentioned above; this is the key fact from elementary (but nontrivial) number theory that any nonzero element of**Z**/*p***Z** has a multiplicative inverse!). This field**Z**/*p***Z** and other prime fields have important uses in number theory, combinatorics, computer science, and elsewhere, often using the linear algebra that we develop in Math 55a.**Z**/*p***Z** - [cf. the boxed note on page 42 of Axler] It is natural to wonder whether
**every**vector space, finite-dimensional or not, has a basis. The polynomial ring*F*[*z*], considered as a vector space over*F*(and denoted by a fancy script*P*in Axler), does have a basis (powers of*z*), as does a polynomial ring in several variables, or even infinitely many (see the next item); but does*F*^{∞}? The answer is yes — but only under the Axiom of Choice (equivalently, Zorn’s Lemma)! [“but only under” because it is known that Choice/Zorn is*equivalent*to the claim that every vector space has a basis. Don’t spend too much time trying to find an explicit basis for*F*^{∞}, or for**R**as a vector space over**Q**(a “Hamel basis”)…] Using the same tool one can prove analogues of some other results in Chapter 2, such as 2.33 (p.41: every linearly independent set extends to a basis), and thus 2.34 (p.42: every subspace is a direct summand; again, don’t spend too much time trying to do this explicitly for**Q**as a subspace of the**Q**-vector space**R**, or for⊕ as a subspace_{n≥1}*F*of !). NB some other results clearly fail in infinite dimensions, even when we have an explicit basis; e.g. the even powers of*F*^{∞}*z*form a linearly independent subset of*F*[*z*] that has the same cardinality as a basis but is not a basis. - However, 2.31 (p.40: every spanning list contains a basis)
still holds with no further axioms for spanning sets
*S*of arbitrary size, as long as*V*is finite dimensional. The reason is that*V*has a finite spanning set, say*S*_{0}, and every element of*S*_{0}is a linear combination of elements of*S*, and sincelinear combinations are of necessity finite it takes only a finite subset of*S*to span*S*_{0}and thus*V*. Now apply the proof of 2.31 to this finite subset. We may call this generalization “2.31+”. - [from the Sep.12 lecture] Here’s an extreme example of how
basic theorems about finite-dimensional vector spaces can become
utterly false for finitely-generated modules:
a module generated by just one element
can have a submodule that is not finitely generated. Indeed,
for any field
*F*let*A*be the ring of polynomials in infinitely many variables*X*. [The letter “_{j}*A*” is a common name for a ring, from French*anneau*, cognate with English “annulus”.] As usual we can regard*A*as a module over itself, with a single generator 1. Then a submodule is just an ideal of the ring. Choose the ideal*I*generated by all the*X*, which consists of all polynomials with constant coefficient equal 0. Then if there are infinitely many indices_{j}*j*then*I*is infinitely generated; indeed any generating set must be at least as large as the index set of , so for every cardinal ℵ we can make a ring*j*’s*A*with a singly-generated module (namely*A*itself) and with a submodule that cannot be generated by fewer than ℵ elements.

For a subtler example, consider the ring we might call “*F*[*X*^{1/2∞}]”, consisting of*F*-linear combinations of monomials*X*^{n/2k}for arbitrary nonnegative integers*n*and*k*. Again let*I*be the ideal generated by the nonconstant monomials, which is not finitely generated, though there are generating sets that are “only” countably infinite. The new behavior involves the countable generating set{ : there is no minimal generating subset, because each*X*^{1/2k}| k≥0} is a multiple of*X*^{1/2k} for any*X*^{1/2k'}k'>k . Likewise for the ring generated by all monomials*X*^{r}with*r*any nonnegative rational number (or even all*X*^{r}with*r*any nonnegative real number).

(When*A*is Noetherian, submodules of finitely-generated modules*are*finitely-generated, but might still require more generators; for example, there are Noetherian rings*A*with “non-principal ideals”*I*, which give examples of a 1-generator module with a submodule that requires at least 2 generators.) - As with the notions of span and linear combination, the definition of
a linear transformation makes sense for modules over any ring
*A*(whether commutative or not), and in that generality is called an (so you now know the “morphisms” in the “category” of*A-module homomorphism* ); when*A*-modules*A*is a skew field, we still call this a linear transformation, and the “rank-nullity theorem” (3.22, page 63) still holds for finite-dimensional vector spaces in that context. - Suppose
*T*:*V*→*W*is a linear transformation. Axler’s notation for the image of*T*was already becoming rather old-fashioned when he wrote the first edition of his book; these days simply is common (and likewise for any function at all). The terminology “null space” (whether one or two words) for*T*(*V*) is also somewhat quaint; we usually say “kernel” and write “ker(*T*^{ −1}({0})*T*)” [and (LA)TeX already provides the command`\ker`to typeset this properly]. While I’m at it, best to avoid the use of“one-to-one” to mean“injective” (see boxed note on page 60), because it is also sometimes used for“bijective” . - Please avoid Axler’s notation “product” and
“ (p.91, 3.71 ff.). I understand the motivation for this notation: it is formally correct, and avoids the need to distinguish between “external direct sum” (the usual name for that vector space) and “internal direct sum” (a vector space sum [within some larger vector space] that happens to be direct). The problem with this is that in Math 55 (and ubiquitously in the literature) we shall introduce before long a “tensor product”*V*×*W*” of vector spaces, whose dimension is the product of the dimensions of*V*⊗*W**V*and*W*when those two dimensions are finite; and it would be a much bigger source of confusion to have*that*notation coexist with“ where the dimensions add. So please stick with*V*×*W*”“ and the name “external direct sum” — or if you must, “Cartesian product” to avoid confusion with tensor products. For a possibly infinite Cartesian product, which is*V*⊕*W*”*not*the same as a direct sum (because an element of the direct sum must have only finitely many nonzero components), we still have the notationΠ to distinguish the Cartesian product from the direct sum_{i∈I}*V*_{i}⊕ ._{i∈I}*V*_{i} - More notes on notation: I understand why Axler wants to distinguish
*V'*and*T'*(dual space and transformation) from*V**and*T**, and*U*^{0}(annihilator) from*U*^{⊥}. (See the boxed note on page 104.) I’ll try to stick with*U*^{0}in this class. But for the duals, using “ ' ” this way incurs a steep price of the very useful construction exemplified by “let*V*and*V'*be vector spaces”: we already have few enough good letters to name mathematical structures that even π is pressed into double duty (not just 3.14159… but also the quotient πrojection from*V*to*V*/*U*). I’ll stick with the common*V**and*T**here. - An equivalent statement of the identity (
*ST*)* =*T*S**(third part of 3.101, page 104 of Axler), together with (*I*)* =_{V}*I*(which Axler might not even bother stating explicitly), is that duality of vector spaces and linear transformations constitutes a “contravariant functor” from the category of_{V*} spaces and linear transformations to itself.*F*-vector - The results about quotient spaces and duality in sections E and F of
Chapter 3 are often described in terms of
*exact sequences*. A sequence… → of linear transformations (or*L*→*M*→*N*→ … homomorphisms, “etc.”) is said to be “exact at*A*-module*M*” if the kernel of the map is the image of the map*M*→*N* (that is, if the elements of*L*→*M**M*that go to zero in*N*are precisely those that come from*L*). The sequence is “exact” if it is exact at each step with both an incoming and an outgoing map. In particular a map is injective*M*→*N*__iff__it extends to a sequence0 → that is exact at*M*→*N**M*, and surjective__iff__it extends to a sequence that is exact at*M*→*N*→ 0*N*. (Note that in each case there is no choice about the function from or to the trivial vector space 0, and likewise at least for modules.) Thus the map is an isomorphism__iff__0 → is exact (at both*M*→*N*→ 0*M*and*N*). Even more easily,0 → is exact*M*→ 0__iff__*M*=0. A*short exact sequence*is the next case, with three modules other than the initial and final 0. The standard example is0 → where the map*L*→*M*→*M*/*L*→ 00 → is an inclusion map (thus an injection) and the map*L*→*M* is the quotient map (thus a surjection). In general if*M*→*M*/*L*→ 00 → is a short exact sequence then the injection*L*→*M*→*N*→ 0 identifies*L*→*M**L*with a submodule of*M*, and then the surjection is identified with the quotient map. More generally,*M*→*N**any*homomorphism extends (uniquely up to equivalence) to an exact sequence with*L*→*M**four*modules between the outer zeros:0 → , where*K*→*L*→*M*→*N*→ 0*K*is the kernel of the map , and*L*→*M**N*is its “*cokernel*”, that is, the quotient of*M*by the image of*L*.Now consider the case of vector spaces. Then to each linear transformation

we associate the dual transformation*V*→*W* , with the dual of a composition*V**←*W** being the composition of the dual transformations*V*→*W*→*X* in reverse order; this makes duality a “contravariant functor” on the category of*V** ←*W** ←*X** spaces. The key fact is that*F*-vector*for finite-dimensional vector spaces, duality preserves exactness*of sequences of linear transformations. Thus starting from any linear , we can extend to an exact sequence*V*→*W*0 → with*U*→*V*→*W*→*X*→ 0*U*the kernel and*X*the cokernel, and dualize to deduce the exactness of0 ← with*U**←*V**←*W**←*X**← 0 the dual map. This immediately encodes Axler 3.108 (page 107): the map*V**←*W** is surjective*V*→*W*__iff__*X*is zero__iff__*X**is zero__iff__the dual map is injective. Likewise for 3.110 (p.108) via the vanishing of*U*and*U**. With a bit more work we can get the general relationsker( (3.107, p.106) and*T**) = (im(*T*))^{0}im( (3.109, p.107) between the kernels and images of*T**) = (ker(*T*))^{0}*T*and its dual, again assuming that*T*is a linear map between finite-dimensional vector spaces. Conversely, the fact that duality preserves exactness (for sequences of linear maps between finite-dimensional vector spaces) can be deduced as a special case of 3.107 and 3.109.You can now understand this joke (such as it is).

- Another way to think about the eigen-basics: “Lemma 5.0”:
If
*T*is an operator on any vector space*V*, and λ any scalar, then*U*is an invariant subspace for*T*__iff__it is an invariant subspace for . So, for instance, since ker*T*− λ*I*_{V}*T*is an invariant subspace, so isker( , a.k.a. the*T*− λ*I*)_{V}λ-eigenspace . - Yet another note on notation: Axler’s name
“
*T/U*” (for the operator on*V/U*induced from the action of*T*on a vector space*V*with an invariant subspace*U*, see 5.14 on p.137) is a nice notation, but (unlike for the restriction of*T*|_{U}*T*to*U*) is seen rarely if at all in the research literature. Normally it will be called plain*T*, or possibly`\bar{T}`(since it is constructed by descending to*V/U*the composition of*T*with the quotient map ).*V*→*V/U* - Let
*T*be a linear operator on*V*. The algebraic properties of polynomial evaluation at*T*can be summarized by saying that the mapfrom *F*[*X*]to End( that takes any polynomial*V*)*P*to is not just linear but a*P*(*T*)*ring homomorphism*. [Since is a commutative ring, so is the image of this homomorphism, even though*F*[*X*]End( is not commutative once*V*)dim( .] In particular the kernel is an ideal*V*) > 1in ; when*F*[*X*]*V*is finite dimensional, this ideal must be nonzero, and its generator is what we shall call the “minimal polynomial” of*T*.*Special case:*if*V*is*F*itself, then we naturally identifyEnd( with*V*)*F*, and we get for any field element*x*the evaluation homomorphismfrom to*F*[*X*]*F*that takes any polynomial to its value at*x*. - Axler proves the Fundamental Theorem of Algebra using complex analysis, which cannot be assumed in Math 55a (we’ll get to it at the end of 55b). Here’s a proof using the topological tools we’ll develop at the start of 55b. (Axler gives one standard complex-analytic proof in 4.13 on page 124.)
- Triangular matrices are intimately related with “flags”.
A (complete)
*flag*in a finite dimensional vector space*V*is a sequence of subspaces {0}=*V*_{0},*V*_{1},*V*_{2}, …,*V*=_{n}*V*, with each*V*of dimension_{i}*i*and containing*V*_{i−1}. A basis determines a flag:*v*_{1},*v*_{2}, …,*v*_{n}*V*is the span of the first_{i}*i*basis vectors. Another basis determines the same flag if and only if each*w*_{1},*w*_{2}, …,*w*_{n}*w*is a linear combination of_{i} (necessarily with nonzero*v*_{1},*v*_{2}, …,*v*_{i}*v*coefficient). The_{i}*standard flag*in*F*is the flag obtained in this way from the standard basis of unit vectors_{n} . The punchline is that, just as a diagonal matrix is one that respects the standard basis (equivalently, the associated decomposition of*e*_{1},*e*_{2}, …,*e*_{n}*V*as a direct sum of 1-dimensional subspaces),*an upper-triangular matrix is one that respects the standard flag.*Note that the*i*-th diagonal entry of a triangular matrix gives the action on the one-dimensional quotient space*V*/_{i}*V*_{i−1}(each*i*=1,…,*n*).

- One of many applications is the
**trace**of an operator on a finite dimensional*F*-vector space*V*. This is a linear map from Hom(*V*,*V*) to*F*. We can define it simply as the composition of two maps: our identification ofHom( with the tensor product of*V*,*V*)*V*^{*}and*V*, and the natural map from this tensor product to*F*coming from the bilinear map taking( to*v*^{*},*v*) . We shall see that this is the same as the classical definition: the trace of*v*^{*}(*v*)*T*is the sum of the diagonal entries of the matrix of*T*with respect to any basis. The coordinate-independent construction via tensor algebra explains why the trace does not change under change of basis. (The invariance can also be proved by checking explicitly that*AB*and*BA*have the same trace for any square matrices of the same size.)*A*,*B* - Here are some basic definitions and facts about general
**norms**on real and complex vector spaces. - Just as we can study bilinear symmetric forms
on a vector space over any field, not just
**R**, we can study sesquilinear conjugate-symmetric forms on a vector space over any field*with a conjugation*, not just**C**. Here a “conjugation” on a field*F*is a field automorphismσ: such that σ is not the identity but σ*F*→*F*^{2}is (that is, σ is an involution). Given a basis {*v*} for_{i}*F*, a sesquilinear form ⟨.,.⟩ on*F*is determined by the field elements and is conjugate-symmetric if and only if*a*_{i,j}= ⟨*v*_{i},*v*_{j}⟩, for all*a*= σ(_{j,i}*a*)_{i,j}*i,j*. Note that the “diagonal entries”*a*— and more generally ⟨_{i,i}*v*,*v*⟩ for any*v*in*V*— must be elements of the subfield of*F*fixed by σ. - “Sylvester’s Law of Inertia” states that
for a nondegenerate pairing on a finite-dimensional vector space
*V*/*F*, where either*F*=**R**and the pairing is bilinear and symmetric, or*F*=**C**and the pairing is sesquilinear and conjugate-symmetric,*the counts of positive and negative inner products for an orthogonal basis constitute an invariant of the pairing*and do not depend on the choice of orthogonal basis. (This invariant is known as the “signature” of the pairing.) The key trick in proving this result is as follows. Suppose*V*is the orthogonal direct sum of subspaces*U*_{1},*U*_{2}for which the pairing is positive definite on*U*_{1}and negative definite on*U*_{2}. Then any subspace*W*of*V*on which the pairing is positive definite has dimension no greater than dim(*U*_{1}). Proof: On the intersection of*W*with*U*_{2}, the pairing is both positive and negative definite; hence that subspace is {0}. The claim follows by a dimension count, and we quickly deduce Sylvester’s Law. - Over any field not of characteristic 2,
we know that for any non-degenerate
symmetric pairing on a finite-dimensional vector space
there is an orthogonal basis, or equivalently
a choice of basis such that the pairing is
(
*x*,*y*)=Σ_{i}(*a*) for some nonzero scalars_{i}x_{i}y_{i}*a*. But in general it can be quite hard to decide whether two different collections of_{i}*a*yield isomorphic pairings. Even over_{i}**Q**the answer is already tricky in dimensions 2 and 3, and I don’t think it’s known in a vector space of arbitrary dimension. Over a finite field of odd size there are always exactly two possibilities, as I think we’ll see in a few weeks. - If
*U*is a subspace of inner-product space*V*, but not necessarily finite dimensional, there is not generally a complement: one can still define*U*^{⊥}, but the direct orthogonal sum might be strictly smaller than*U*⊕*U*^{⊥}*V*. What then happens to6.56 (p.198 in 6.C) , which describes the orthogonal projection as the vector in*P*(_{U}*v*)*U*closest to*v*(i.e., minimizing the norm|| )? Well,*v*−*u*||*if*there exists such*u*then indeed is orthogonal to*v*−*u**U*, but in general the minimum need not be attained: at best we can construct a sequence of vectors*u*in_{n}*U*such that|| *v*−*u*||_{n}*approaches*inf . It then follows from Apollonius’ theorem (see the front cover of Axler! and also Exercise 31 of 6.A, page 179) that the_{u∈U}||*v*−*u*||*u*constitute a Cauchy sequence in_{n}*U*(else( is too close to*u*+_{m}*u*)/2_{n}*v*). So if*U*is*complete*with respect to the norm distance then there is a nearest vector and we can proceed as before. But in general infinite-dimensional inner product spaces are not complete (the complete ones are Hilbert spaces, and that is a very special case). We shall say a lot more about completeness and related notions at the start of Math 55b. - A regular graph of degree
*d*is a*Moore graph*of girth 5 if any two different vertices are linked by a unique path of length at most 2. Such a graph necessarily has vertices. Let*n*= 1 +*d*+*d*(*d*−1) =*d*^{2}+ 1*A*be the adjacency matrix, and**1**the all-ones vector. Then (because each vertex has degree*d*)**1**is a of*d*-eigenvector*A*. We have(1 + for all*A*+*A*^{2})*v*=*d v*+ ⟨*v*,**1**⟩**1***v*(proof: check on unit vectors and use linearity). Thus*A*takes the orthogonal complementof to itself and satisfies**R**·**1**(1 + on that space. Since this quadratic equation has distinct roots*A*+*A*^{2}) =*d**m*and−1− for some*m**m*> 0(1 + ), it follows that the orthogonal complement*m*+*m*^{2}) =*d*of is the direct sum of the corresponding eigenspaces. Let**R**·**1***d*_{1}and*d*_{2}be their dimensions. These sum to , and satisfy*n*−1 =*d*^{2} because the matrix*md*_{1}+ (−1−*m*)*d*_{2}+*d*= 0*A*has trace zero. This lets us solve for*d*_{1}and*d*_{2}. in particular we find that their difference is(2 . Since that’s an integer, either*d*−*d*^{2}) / (2*m*+1) (giving the pentagon graph) or*d*=2*m*is an integer. Substituting for*m*^{2}+*m*+ 1*d*, we find that16( is an integer plus*d*_{1}−*d*_{2})15/(2 , whence*m*+1)*m*is one of 0, 1, 2, or 7. The first of these is impossible, and the others give , 7, or 57 as claimed.*d*= 3It is clear that the pentagon is the unique Moore graph of girth 5, and degree 2, and it is not hard to check that the Petersen graph is the unique example with

. For*d*= 3 there is again a unique graph up to an interesting group of automorphisms; see this proof outline, which also lets one count the automorphisms of this graph.*d*= 7 - Why the name “spectral theorem”? The set (or sometimes the
“multiset”) of eigenvalues
of a linear operator on a vector space
*V*is often called its “spectrum”, especially when*V*is a real or complex vector space, either finite or infinite dimensional. This is related with the visual (and by extension the electromagnetic) spectrum, for reasons that would take us much too far into wave and quantum mechanics, so we shall say little more of that here (but you may encounter it again in your physics class(es)).

Interlude: normal subgroups;
short exact sequences in the context of groups:
A subgroup *H* of *G* is **normal**
(satisfies *H* = *g*^{−1}*Hg**g* in *G*) __iff__
*H* is the kernel of some group homomorphism from *G*
__iff__ the injection *H* → *G* *H*
→ *G*
→ *Q*
→ {1},
*Q* is the **quotient group**
*G*/*H**H*
→ *G*
→ *Q*
→ 1.]
*A _{n}*
→

Some more tidbits about exterior algebra:

- If
*w*,*w*' are elements of the*m*-th and*m*'-th exterior powers of*V*, then*ww*'=(−1)^{mm'}*w*'*w*; that is,*w*and*w*' commute unless*m*and*m*' are both odd in which case they anticommute. - If
*m*+*m*'=*n*=dim(*V*) then the natural pairing from the*m*-th and*m*'-th exterior powers to the*n*-th is nondegenerate, and so identifies the exterior power canonically with the dual of the*m*'-th*m*-th*tensored with the top (n-th) exterior power*. - In particular, if
*m*=1, and*T*is any invertible operator on*V*, then we find that the induced action of*T*on the (*n*−1)st exterior power is the same as its action on*V*^{*}multiplied by det(*T*). This yields the formula connecting the inverse and cofactor matrix of an invertible matrix (a formula which you may also know in the guise of “Cramer’s rule”). - For each
*m*there is a natural non-degenerate pairing between the*m*-th exterior powers of*V*and*V*^{*}, which identifies these exterior powers with each other’s dual.

We’ll also show that a symmetric (or Hermitian) matrix
is positive definite __iff__ all its eigenvalues are positive
__iff__ it has positive principal minors
(the “principal minors” are the determinants
of the square submatrices of all orders containing the (1,1) entry).
More generally we’ll show that the eigenvalue signs determine
the signature, as does the sequence of signs of principal minors
if they are all nonzero. More precisely: an invertible
symmetric/Hermitian matrix has signature
*r*,*s*)*r* is the number of positive eigenvalues and
*s* is the number of negative eigenvalues;
if its principal minors are all nonzero then
*r* is the number of *j* in
*n*}*j*-th*j*−1)-st*s* is the number of *j* in that range such that the
*j*-th*j*−1)-st*j*=1^{s}*r*',*s*')*r*' ≤ *r**s*' ≤ *s*

For positive definiteness, we have the two further
equivalent conditions: the symmetric (or Hermitian) matrix
*A*=(*a _{ij}*) is positive definite

For any matrix

(1) There exists a nonnegative integer

(2) For any vector

(3)

Note that (1) and (2) make no mention of the dimension, but are still not equivalent for operators on infinite-dimensional spaces. (For example, consider differentiation on the

(4) There exists a basis for

(5) Every upper-triangular matrix for

Recall that the second part of (5) is automatic if

First problem set / Linear Algebra I: vector space basics; an introduction to convolution rings

• “Which if any of these basic results would fail if

• Problem 12: If you see how to compute this efficiently but not what this has to do with Problem 8, please keep looking for the connection.

Here’s the “Proof of Concept” mini-crossword with links concerning the ∎ symbol. Here’s an excessively annotated solution.

Second problem set / Linear Algebra II:
dimension of vector spaces; torsion groups/modules and divisible groups

**About Problem 5:** You may wonder: if not determinants,
what *can* you use? See Axler, Chapter 4, namely 4.8 through 4.12
*infinite*
field; Axler’s proof is special to the real and complex numbers,
but 4.12 yields the result in general. (We already remarked that
this result does not hold for finite fields.)

Third problem set / Linear Algebra III: Countable vs. uncountable dimension of vector spaces; linear transformations and duality

Fourth problem set / Linear Algebra IV: Duality; projective spaces; more connections with polynomials

Fifth problem set / Linear Algebra V:
“Eigenstuff”
(with a prelude on exact sequences and more duality)

**corrected** 4.x.10:

*Problem 2:* *F ^{N}* /

Sixth problem set / Linear Algebra VI:
Tensors, eigenstuff cont’d, and a bit on inner products

Seventh problem set / Linear Algebra VII: Pairings and inner products, cont’d

Eighth problem set / Linear Algebra VIII: The spectral theorem; spectral graph theory; symplectic structures

Ninth problem set / Linear Algebra IX: Trace, determinant, and more exterior algebra

Tenth problem set: Linear Algebra X (determinants and distances); representations of finite abelian groups (Discrete Fourier transform)

Eleventh and final problem set: Representations of finite abelian groups