Tentative schedule and topics covered

  • Lecture 1 (Thursday, 9/26)- Introduction.  Section 1.1.  Some basics of matrix multiplication, identity matrices, inverse matrices, associativity.  Most classes will be more straightforward and will jump around less (but will have harder material).
  • Lecture 2 (Friday, 9/27)- Homework 1 distributed.  Time in class to work on it in groups.
  • Lecture 3 (Monday, 9/30)- Sections 1.2.  Proved carefully that a square matrix with a one-sided inverse is invertible, using row operations.  Gave an example of how this is not true for non-square matrices.  In general quickly discussed row operations, elementary matrices, and row echelon form.  Mentioned "matrix units" e_{i,j}.
  • Lecture 4 (Tuesday, 10/1)- Discussion section.
  • Lecture 5 (Wednesday, 10/2)- Section 1.5.  Permutations.  Defined permutations and transpositions, the order of a permutation, and described why every permutation of a finite set can be expressed as a finite composition of transpositions.  Defined the notion of "even" and "odd" permutations.  Introduced cycle notation and stated that if the cycles are disjoint, then the order is the lcm of the cycle lengths.
  • Lecture 6 (Thursday, 10/3)- Sections 1.4 and 1.6.  Determinants.  Used the characterizing properties of determinants from Theorem 1.4.7 to derive lots of consequences, including that a matrix is invertible if and only if its determinant is non-zero, and multiplicativity of the determinant.
  • Lecture 7 (Friday, 10/4)- Homework 1 due.  Homework 2 distributed.  Time in class to work on it in groups.
  • Week 2- This week I would like to cover material from 2.1-2.6, 2.9, 2.11.
  • Lecture 8 (Monday, 10/7)-  Start of Chapter 2.  Lots of definitions and examples and non-examples: Binary operations, associative, identity, inverses, groups, abelian groups, subgroups.  Proved that every subgroup of (Z,+) has the form mZ for some integer m.
  • Lecture 9 (Wednesday, 10/9)- Lots of definitions: order of an element, cyclic groups, the subgroup generated by a set, product groups, homomorphism, isomorphism, kernel.  Proved carefully that Q and Z are not isomorphic (with group operation addition).
  • Lecture 10 (Thursday, 10/10)- Discussed solutions to i, iii, v, xv, xx, xxi from this Download Groups Worksheet.
  • Lecture 11 (Monday, 10/14)- Defined cosets, the index, and normal subgroups.  Proved Lagrange's theorem and that every group of prime order is cyclic.  Checked that the kernel of a homomorphism is a normal subgroup of the domain.
  • Lecture 12 (Wednesday, 10/16)- Briefly introduced quotient groups and the 1st isomorphism theorem, then did an activity involving finding kernels and images of homomorphisms.  Here are Download solutions.
  • Lecture 13 (Thursday, 10/17)- Sketched the proof of the 1st isomorphism theorem.  Proved that no quotient of S_3 could be cyclic of order 3 by saying if (S_3)/K is order 3, then every transposition is in K.  Gave equivalent characterizations of a subgroup being normal, and proved that every index 2 subgroup is normal.
  • Lecture 14 (Monday, 10/21)- Started Chapter 3. Very quick introduction to fields (mostly just examples of fields), and then an introduction to vector spaces, including subspace, span, and linear independence.
  • Lecture 15 (Wednesday, 10/23)- Proved that a maximal linearly independent set is a basis.  Used Theorem 3.4.18 in Artin to prove that every vector space spanned by a finite set of vectors has a basis, and that two finite bases have the same cardinality.  Started to introduce Zorn's lemma by talking about partially ordered sets and totally ordered sets.  Stated the Axiom of Choice but not Zorn's lemma.
  • Lecture 16 (Thursday, 10/24)- Used Zorn's lemma to prove that every vector space has a basis, and briefly introduced the notion of a direct sum of finitely many subspaces of a vector space (Section 3.6).
  • Lecture 17 (Monday, 10/28)- Chapters 4.1 and 4.2.  Defined linear transformations, kernel, image, nullity, rank, isomorphism of vector spaces.  Proved the rank nullity theorem.  Discussed how a linear transformation corresponds to a matrix, once ordered bases are chosen.
  • Lecture 18 (Wednesday, 10/30)- Discussed change of basis in general (not for matrices) and proved Theorem 4.2.10a.
  • Lecture 19 (Thursday, 10/31)- Discussed change of basis for permutations (e.g., conjugate permutations have the same cycle type), for vector spaces (section 3.5), and for matrices (section 4.3).
  • Lecture 20 (Friday, 11/1)- Discussed questions (iv), (v), (vii), (ix), (x), (xi), (xiii), (xv) from this Download midterm review worksheet.
  • Lecture 21 (Tuesday, 11/5)- Discussed eigenvectors and eigenvalues (Section 4.4) and proved Proposition 4.6.5.
  • Lecture 22 (Wednesday, 11/6)- Section 4.5: Characteristic polynomial.
  • Lecture 23 (Thursday, 11/7)- Sections 4.6 and 4.7.  The correspondence between diagonal matrices and bases of eigenvalues.  Every square matrix over C is similar to an upper-triangular matrix.  Two matrices with the same characteristic polynomial and which are not similar.  Introduction to Jordan form (including uniqueness), without proof.
  • Lecture 24 (Wednesday, 11/13)- Section 5.2.  Proved the Cayley-Hamilton theorem over C using the following outline.  1.  It is true for diagonal matrices.  2.  It is true for diagonalizable matrices.  3.  Every matrix is a limit of diagonalizable matrices.  4.  The set of matrices for which Cayley-Hamilton is correct is both a dense set (by 3) and a closed set (because it can be expressed in terms of vanishing of polynomials), hence Cayley-Hamilton is valid for all complex matrices.
  • Lecture 25 (Thursday, 11/14)- Section 5.1.  Discussed basic properties of orthogonal matrices and orthogonal operators.  Discussed the 2x2 case.  (We won't discuss the 3x3 case, which is emphasized in Artin.)
  • Lecture 26 (Monday, 11/18)- Section 6.2.  Defined isometries and discussed their basic properties.  Stated without proof that an isometry which fixes 0 is an orthogonal operator, and that every isometry is a composition of first an orthogonal operator and then a translation.
  • Lecture 27 (Wednesday, 11/20)- Sections 6.7 and 6.4.  Briefly introduced group actions and the notions of orbit and transitive.  Then discussed different ways to think about the dihedral group.
  • Lecture 28 (Thursday, 11/21)- Continued discussion of dihedral groups.  Applications using the generators and relations for a dihedral group.  The center of the dihedral group.  The fact that the dihedral group has many elements of order 2.  Proving that certain familiar groups are not isomorphic to dihedral groups (S_n for n >= 4, A_4, GL_2(Z/pZ) for p >= 5).
  • Lecture 29 (Monday, 11/25)- The orbit-stabilizer theorem and some of its consequences.  We used it to prove that the center of a p-group is non-trivial.  The orbit-stabilizer theorem is Proposition 6.8.4 in the textbook.
  • Lecture 30 (Wednesday, 11/27)- Proved that every group of order p^2 is abelian.  Described how generators and relations can be used to define maps from a group to another group, and showed as an example that there is an isomorphism from D_3 to S_3.  Gave an example of a non-abelian group of order 27 in which all the non-trivial elements have order 3.
  • Lecture 31 (Monday, 12/2)- Discussed the group GL_n(F).  Has a subgroup isomorphic to F - {0} and another subgroup isomorphic to F (if n is at least 2).  It is non-abelian if n is at least 2.  We computed the cardinality of GL_1(Z/pZ), GL_2(Z/pZ), and GL_3(Z/pZ).  We gave an injective group homomorphism from S_n to GL_n(F), and used it to give another characterization of the alternating group A_n.
  • Lecture 32 (Wednesday, 12/4)- Review.  Discussed dihedral groups and Jordan form, including the definition of generalized eigenvectors.
  • Lecture 33 (Thursday, 12/5)- Review.  Discussed (without proof) the claim that if T: V->V is a diagonalizable linear operator on a finite-dimensional vector space, then the restriction of T to any T-invariant subspace of V is also diagonalizable.  Used the contrapositive of this to prove that if there is a generalized eigenvector which isn't an eigenvector, then the linear operator is not diagonalizable.  Then discussed questions (iii), (iv), (vi) in the Download review worksheet.
  • Lecture 34 (Friday, 12/6)- Review.  Discussed questions (i), (ii), (vii), (viii), (ix), (xi), (xii) from Download review worksheet 2.  Our answer for (ii) involved the following trick: If G is an abelian group and all non-identity elements have order p (p prime), then G is isomorphic to (V,+), where V is some vector space over Z/pZ.