Material covered on Tuesday January 15: orthogonal projections (see notes above); adjoints; Theorem 6.10; easy direction of proofs of Theorems 6.16 and 6.17.

Material covered on Thursday January 17: Theorems 6.8 and 6.9, properties of adjoints.

Material covered on Tuesday January 22:

Theorem 6.15

Example of T not normal with x an eigenvector of T but x not an eigenvector of the adjoint of T.

Example of a normal operator T on a real inner product space, with T having no eigenvalues.

Using Schur's Lemma (Theorem 6.14) to prove Theorem 6.16.

Example: If dim(V) is finite and F is the complex numbers, T is linear and TT=T, prove T is self adjoint if and only if T is normal.

Material covered on Thursday January 24:

Example: Show that T(adjoint T)=UU for some self adjoint U (finite-dimensional case).

Explanation of how Schur's lemma is proved.

Material covered on Tuesday January 29:

Translating Theorem 6.16 into a theorem about complex matrices (Theorem 6.19).

Lemma: If Q is a change of coordinate matrix relative to two orthonormal bases, then Q* is the inverse of Q.

Example: Given a specific 4x4 normal complex matrix A, find a diagonal D and a unitary Q with D=QAQ*.

Eigenvalues of self adjoint operators are real; Proving Theorem 6.17.

Translating Theorem 6.17 into a theorem about real matrices (Theorem 6.20).

Example of a self adjoint linear operator with no eigenvalues (infinite dimensional).

Definition of unitary matrix, orthogonal matrix, unitary linear operator, orthogonal linear operator.

Material covered on Thursday January 31:

AA*=A*A=I if and only if the rows (columns) of A form an orthornormal set relative to the standard inner product.

Basic properties of unitary and orthogonal operators (Theorem 6.18).

Corollary 1 and 2 from Section 6.5.

Material covered on Tuesday February 5:

Several examples involving unitary and orthogonal operators.

General comments about the spectral theorem.

Material covered on Thursday February 7:

T-invariant subspaces; cyclic subspace generated by a vector;

Examples of T-invariant subspaces; Theorem 5.21;

Statement of Cayley-Hamilton Theorem.

Material covered on Thursday February 14:

Cayley-Hamilton Theorem for T diagonalizable.

Theorem 5.22.

Proof of the Cayley-Hamilton theorem (Theorem 5.23).

Material covered on Thursday February 28:

Minimal polynomials; Theorem 7.12, Theorem 7.14.

Material covered on Tuesday March 4:

Examples related to minimal polynomial material.

Minimal polynomial of a diagonalizable linear operator.

General comments about Jordan canonical form.

Definition of generalized eigenspace of a linear operator.

Theorem 7.1.

Definition of cycle of generalized eigenvectors of a linear operator.

Material covered on Thursday March 6:

Theorem 7.2, Theorem 7.3, Theorem 7.4 part 3.

Material covered on Tuesday March 11:

More about Theorems 7.2, 7.3 and 7.4.

Example of computing a Jordan basis and Jordan form.

Theorem 7.6.

Material covered on Thursday March 13:

Dot diagrams; Theorem 7.9; uniqueness of Jordan canonical form.

Example: computing dot diagrams, Jordan canonical form, minimal polynomial.

Material covered on Tuesday March 18:

Theorem 7.16 in the case where the characteristic polynomial splits over F.

Relation between Jordan canonical form and minimal polynomial (see question 3 on problem set 6).

Several examples.

Proof of Theorem 7.7.

Material covered on Thursday March 20:

Similarity of matrices and Jordan canonical form.

Dual spaces (examples; Theorem 2.24; statement of Theorem 2.26).

Material covered on Tuesday March 25:

Theorem 2.26 and Theorem 2.25.

Definition of group; subgroups; examples.

Material covered on Thursday March 27:

Definition of group homomorphism (p.87); kernel and image.

Definition of group isomorphism (p.32).

Defn: A subgroup H of G is normal if the product gx(inv(g)) belongs to H for all x in H and g in G.

(Here, inv(g) is the inverse of g.)

If f is a group homomorphism from G to G', then f(e)=e',

f(inv(x))=inv(f(x)) for x in G; ker f is a normal subgroup of G;

ker f ={e} if and only if f one-to-one; f(G) is a subgroup of G'.

Examples of homomorphisms.

Material covered on Tuesday April 1:

Definition of rotation, definition of reflection pp.472-473,FIS, or Chapter 9 of Armstrong.

A 2 by 2 orthogonal matrix is a reflection if and only if det A=1.

A 2 by 2 orthogonal matrix is a rotation if and only if det A=-1.

(n>2) Dihedral group D_n - symmetry group of a regular n-gon (Chapter 4, Armstrong)

Realizing elements of D_n as 2 by 2 orthogonal matrices; rotations in D_n; reflections in D_n;

Order of an element of a group (p.18).

Definition of cyclic subgroup generated by one element of a group (p.21);

The group of rotations in D_n is a cyclic normal subgroup of D_n;

The subgroup of D_n generated by a reflection has order 2, and is not a normal subgroup.

Material covered on Thursday April 3:

Definition of the subgroup of a group G generated by a subset of G.

If x in G has order n, and f is a homomorphism from G to G', then the order of f(x) divides n.

If f is one-to-one, then f(x) has the same order as x.

If G and G' are finite groups of the same order, then f is an isomorphism if and only if kernel of f is {e}.

If f is a homomorphism from D_n to G', then f(s)f(s)=e' and f(s)f(r)f(s)=inverse of f(r).

If G is abelian and G' is nonabelian, then G and G' are not isomorphic.

Material covered on Tuesday April 8:

Lagrange's theorem.

Comments about 3 by 3 orthogonal groups.

Permutation groups; cycles; disjoint cycles commute.

Expressing elements of permutation groups as products of disjoint cycles.

The set of tranpositions in S_n generates S_n.

Material for Thursday April 10:

More on permutation groups; alternating groups.

Expressing dihedral groups as subgroups of permutation groups.

Homomorphisms and permutation groups.