Assignments are listed by the date due.
Submitted assignments must be typeset.
Mon., Dec. 16: 
Final Examination 10:30  12:30

Wed., Dec. 11: 
Written assignment no. 5 (also available as PDF or DVI) is due.
Review: Bring questions about the course.

Mon., Dec. 9: 
In class: the notion of endomorphism associated with an n X n matrix of
polynomials. Dimension of this endomorphism is the degree of the
determinant of the matrix of polynomials when the determinant is nonzero.
Every finitedimensional endomorphism is the direct sum of endomorphisms
associated with polynomials and every square matrix is similar to a
direct sum of companion matrices.

Fri., Dec. 6: 
In class: the idea of diagonalizing the characteristic matrix (a matrix
of polynomials) of a given matrix in order to determine the isomorphism
class of the endomorphism associated with the given matrix.

Wed., Dec. 4: 
In class: the notion of linear map from one endomorphism to another.
Isomorphism of endomorphisms and similarity of matrices.

Mon., Dec. 2: 
Written assignment no. 4 (also available as PDF or DVI) is due.
In class: the canonical endomorphism of F[t]/P(t)F[t] when
P is a polynomial.

Fri., Nov. 22: 
In class: the exterior powers of a finitedimensional vector space.

Wed., Nov. 20: 
In class: how to “compute” an alternating plinear form.

Mon., Nov. 18: 
In class: the symmetric powers of a finitedimensional vector space.

Fri., Nov. 15: 
In class: construction of the tensor product.

Wed., Nov. 13: 
Written assignment no. 3 (also available as PDF or DVI) is due.
In class: the abstract tensor product is unique up to unique isomorphism.

Mon., Nov. 11: 
In class: introduction to the abstract notion of tensor product of
two vector spaces.

Fri., Nov. 8: 
In class: Topological decomposition of the real general linear group.

Wed., Nov. 6: 
In class: orthogonal diagonalization of selfadjoint real linear
endomorphisms.

Mon., Nov. 4: 
Written Assignment No. 3 (also available as PDF or DVI) was distributed.
In class: the adjoint of a linear endomorphism relative to a dualizing
bilinear form.

Fri., Nov. 1: 
In class: orthogonal transformations relative to a dualizing bilinear
form; inner products on real vector spaces.

Wed., Oct. 30: 
In class: matricial reformulation of the classification of symmetric
and alternating bilinear forms.

Mon., Oct. 28: 
In class: classification of symmetric and alternating bilinear forms
on finitedimensional vector spaces.

Fri., Oct. 25: 
In class: nondegenerate and dualizing bilinear forms.

Wed., Oct. 23: 
In class: comments on the midterm test; introduction to bilinear maps,
symmetric and alternating bilinear maps.

Mon., Oct. 21: 
Midterm Test during the class hour.

Fri., Oct. 18: 
MidTerm WarmUp Exercises (also available as PDF or DVI)  not for submission.
In class: review.

Wed., Oct. 16: 
Written Assignment No. 2 (also available as PDF or DVI) is due.
In class: annihilators and preannihilators.

Mon., Oct. 14: 
Exercise:
In class: the dual of a vector space.

Fri., Oct. 11: 
In class: the sum of a collection of subspaces of a vector space;
sum versus direct sum.

Wed., Oct. 9: 
Written Assignment No. 2 (also available as PDF or DVI) was distributed.
In class: The (possibly infinite) direct sum and its universal mapping
property.

Mon., Oct. 7: 
Exercises:
In class: The (possibly infinite) direct product and its universal
mapping property.

Fri., Oct. 4: 
Exercises:
In class: the dimension formula; a linear map between finite
dimensional vector spaces of the same dimension is injective if and
only if it is surjective; examples of quotients; the universal
mapping property for F^<S>.

Wed., Oct. 2: 
The first written assignment (also available as PDF or DVI) is due.
In class: corollaries of the universal mapping property for
the quotient construction; dimension of the quotient of a
finitedimensional vector space.

Mon., Sep. 30: 
In class: the quotient of a vector space by a subspace; the universal
mapping property of the quotient.

Fri., Sep. 27: 
In class: second approach to the basis of a linear map with respect
to a pair of bases; the effect of change of bases on the matrix of
a linear map.

Wed., Sep. 25: 
Note: Some of the informal exercises that are assigned may
subsequently appear in an assignment that is to be written out
and submitted. While the agenda for a written assignment may
not be discussed, the discussion of informal exercises is encouraged.
Therefore, for the informal exercises that appear the
window of opportunity for discussion may be suddenly closed by the
inclusion of such an exercise in a written assignment. Prove the following: Proposition: If two matrices of the same size over a field are both in reduced row echelon form and are also row equivalent, i.e., each may be obtained from the other by a finite sequence of elementary row operations, then they must be equal.
In class: written assignment due October 2 was distributed;
the vector space F[t] of polynomials over a field F;
the vector space Hom(V, W) of linear maps from V to W; every linear
map f from F^n to F^m has the form x > Mx for some unique m x n
matrix M; first approach to the matrix of a linear map from V to W
relative to given bases of V and W.

Mon., Sep. 23: 
Make sure that you know how to do all of the review exercises. Prove the following: Proposition: Let S be any set and F any field. Recall that F^S denotes the set of all maps from S to F . Show that a sequence f_1, f_2, . . ., f_N in F^S is linearly independent if and only if there exist points x_1, x_2, . . ., x_N in S such that the determinant of the N X N matrix (f_i(x_j)) is nonzero.
In class: any two bases of a finitely spanned vector space
have the same number of elements proved using row operations;
dimension of a vector space as the number of elements in any basis.

Fri., Sep. 20:  Continue your review of elementary linear algebra,
and do these review exercises (also available as PDF or DVI).
In class: basis of a vector space as a minimal spanning set;
review of basic facts about elementary row operations.

Wed., Sep. 18: 
Study the handout on matrices and bases (also available as PDF or DVI).
In class: linear independence of a subset of
a vector space; basis of a vector space as a maximal linearly independent
set; Zorn's lemma; existence of a basis for any vector space modulo
Zorn's lemma.

Mon., Sep. 16:  Recess: no class. 
Fri., Sep. 13:  Continue with the review exercises.
In class: the linear map given by a matrix; the subspace
spanned by a finite sequence; linear independence of a finite
sequence; the subspace spanned by a subset of a vector space.

Wed., Sep. 11:  Continue your review of elementary linear algebra,
and do the distributed review exercises (also available as PDF or DVI).
In class: the formal definition of vector space over a field,
with a list of five examples. Also: definition of linear map.

Mon., Sep. 9:  Review your knowledge of elementary linear algebra.
In class: the formal definition of field. 
Fri., Sep. 6:  Recess: no class 