189-251B: Honors Linear Algebra

Blog

Week 1. January 6-10.
This week was devoted to the basic definitions and properties of abstract vector spaces over a field F, and of the linear transformations between them. We discussed what it means for a set of vectors to span a vector space, to be linearly independent, and to be a basis for the vector space. This led us to the notion of dimension of a finite-dimensional vector space.

Next week, we will discuss the final exam of last semester, and I will give some solutions to the problems there. I advise that you have a look at the final exam before then, in order to refresh your memory about the questions whose solutions I will be presenting.

Week 2. January 13-17.
Monday and Wednesday saw the proof of basic facts concerning linear independence of vectors and spanning sets. A basis of a vector space V was defined to be a minimal spanning set, or, equivalently, a maximal linearly independent subset, or, equivalently, a subset B of V such that every vector in V can be expressed uniquely as a (finite, by definition) linear combination of the vectors in B. A vector space which has a finite basis is called finite-dimensional. We proved that the cardinality of a basis for a given finite-dimensional vector space V depends only on V and not on the choice of basis. This cardinality is therefore an invariant of V caled its dimension.

We proved various useful facts about bases and linearly independent sets, based on the slightly technical but useful Steinitz substittion lemma. One of the particularly useful corollaries is that every linearly independent subset of a finite-dimensional vector space can be completed to a basis for V (and hence, its cardinality is less than the dimension of V. This flexibility in the choice of basis was a key ingredient in the rank-nullity theorem proved on Wednesday, which, given a linear transformation defined on a finite-dimensional vector space V, relates the dimensions of its image and kernel to the dimension of V.

Friday's lecture was devoted to a quick review of the questions in the final exam from last semester.

Week 3. January 20-24.
This week was devoted to some of the basic material covered in chapter 4 of the on-line notes, most notably the idea of representing vectors in finite dimensional vector spaces by coordinates, and linear transformations between such spaces as matrices, after a suitable choice of bases of the abstract vector spaces involved. We dicussed quotients, the isomorphism theorem for vector spaces, and the theorem about the dimension of the kernel and image. Hopefully this discussion left you with a feeling of deja vu from our very similar treatment of groups last semester -- and if not, go ccarefully through sections 4.2. and 4.3. of the notes.
We illustrated some of the concrete applications of the material seen in class by discussing linear codes, which are simply subspaces of F₂ⁿ, viewed as vector spaces over the field {0,1} with two elements. Friday's lecture concluded with the construction of the remarkable Hamming code, the (essentially unique) error correcting code of length 7 and dimension 4.

Week 4. January 27-31.
This week was devoted to the study of a single linear transformation T from a vector space V to itself. We saw that one can associate to such a T its minimal polynomial, and we proved that, when V has dimension n, the degree of this minimal polynomial is at most n. We also defined eigenvalues and eigenvectors, and proved that the set of eigenvalues of a linear transformation is equal to the set of roots of its minimal polynomial.

Week 5. February 3-7.
We continued our discussion of eigenvalues, eigenvectors, and diagonalisability and its connection with the characteristic polynomial. The most important theorem we proved this week is the primary decomposition theorem which asserts that a vector space V on which a linear transformation T is defined can be broken up into a direct sum of T-stable subspaces indexed by the irreducible factors of the minimal polynomial. If p(x) is such an irreducible factor, and p(x)^e divides the minimal polynomial exactly, then the restriction of T to the associated stable subspace has p(x)^e as its minimal polynomial. From this we deduced that a linear transformation is diagonalisable if and only if its minimal polynomial factors into a product of distinct linear factors.

Week 6. February 10-14.
This week was devoted to a discussion of determinants, which were treated somewhat differently from the on-line notes. By way of motivation, we started with a discussion of the notion of volume on the real vector space Rⁿ. (This volume is usually called the length when n=1, the area when n=2, the volume when n=3, and continues to be designated as a volume for all larger values of n; this is just for lack of further words in the english vocabulary, and the n-dimensional volume differs just as much from its three-dimensional avatar as the surface area does from the length....

We saw that the function which to (v₁, ..., v_n) associates the signed volume of the n-dimensional parallelopiped spanned by these n vectors satisfies two properties: it is multilinear in the v_i, and it is alternating. Remarkably, these two properties are purely algebraic and make sense over an arbitrary field F; indulging in the mathematician's taste for abstraction, a volume function on an n-dimensional vector space over F can just be defined to be a multilinear, alternating function from Vⁿ to F.

Even more remarkably, such volume functions are essentially unique: any two multlinear alternating functions in n variables on an n-dimensional vector space just differ by a constant of proportionality. (This important uniqueness property was proved in class by induction.)

The determinant of a transformation T: V --> V can then be defined as the ratio between the volume function D(T v₁,...,T v_n) and D( v₁,...,v_n), for any non-zero volume function D. In effect, the determinant of T represents the amount by which T distorts (any) volume on V.

We used this somewhat abstract definition of the determinant to desribe the well-known algorithm for computing it via row-reduction, and gave a closed formula for the determinant of a matrix in terms of its entries, as a sum of n! terms indexed by the elements of the symmetric group on n letters.

With the determinant behind our belt, we were finally able to define the characteristic polynomial of a linear transformation T. This polynomial, denoted f_T(x), is simply the determinant of the transformation x I - T, where I denotes the identity tansformation on V. The explicit formula for the determinant of a matrix shows that this function of x is in fact a polynomial in x whose degree is always equal to n=dim(V).