Math 424/524 Assignments

Fall Semester, 2002

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 non-zero. Every finite-dimensional 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 finite-dimensional vector space.

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

Mon., Nov. 18: In class: the symmetric powers of a finite-dimensional 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 self-adjoint 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 re-formulation of the classification of symmetric and alternating bilinear forms.

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

Fri., Oct. 25: In class: non-degenerate 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 pre-annihilators.

Mon., Oct. 14: Exercise:

If U_1 and U_2 are subspaces of a vector space V, prove that

(U_1 + U_2)/U_1 is isomorphic to U_2/(U_1 CAP U_2)

where U_1 CAP U_2 denotes the intersection of U_1 and U_2.

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:

If V and W are vector spaces over F and V X W is their Cartesian product, show that the quotient of V X W by its linear subspace V X (0) is isomorphic to W and the quotient of V X W by its subspace (0) X W is isomorphic to V.

If V and W are vector spaces over F and X is any vector space having a linear subspace V' isomorphic to V for which the quotient X/V' is isomorphic to W, is it necessarily true that X is isomorphic to V X W?

In class: The (possibly infinite) direct product and its universal mapping property.

Fri., Oct. 4: Exercises:

If F is a field, let F[t] denote the vector space of polynomials in the formal variable t over F, and let U_a denote the subspace of all polynomials that are (polynomial) multiples of the linear polynomial t - a, when a is given. Does the isomorphism class of the quotient space F[t]/U_a depend on a?

Let F be a field and S a set. Recall that F^S denote the vector space (over F) of all maps from S to F. Let U denote the subspace F^<S> consisting of all maps from S to F with finite support. How might the quotient vector space F^S/F^<S> be described intuitively to someone who is not familiar with the quotient construction?

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 finite-dimensional 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 non-zero.
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

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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 non-zero. Every finite-dimensional 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 finite-dimensional vector space.
Wed., Nov. 20:	In class: how to “compute” an alternating p-linear form.
Mon., Nov. 18:	In class: the symmetric powers of a finite-dimensional 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 self-adjoint 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 re-formulation of the classification of symmetric and alternating bilinear forms.
Mon., Oct. 28:	In class: classification of symmetric and alternating bilinear forms on finite-dimensional vector spaces.
Fri., Oct. 25:	In class: non-degenerate 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 pre-annihilators.
Mon., Oct. 14:	Exercise: If U_1 and U_2 are subspaces of a vector space V, prove that (U_1 + U_2)/U_1 is isomorphic to U_2/(U_1 CAP U_2) where U_1 CAP U_2 denotes the intersection of U_1 and U_2. 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: If V and W are vector spaces over F and V X W is their Cartesian product, show that the quotient of V X W by its linear subspace V X (0) is isomorphic to W and the quotient of V X W by its subspace (0) X W is isomorphic to V. If V and W are vector spaces over F and X is any vector space having a linear subspace V' isomorphic to V for which the quotient X/V' is isomorphic to W, is it necessarily true that X is isomorphic to V X W? In class: The (possibly infinite) direct product and its universal mapping property.
Fri., Oct. 4:	Exercises: If F is a field, let F[t] denote the vector space of polynomials in the formal variable t over F, and let U_a denote the subspace of all polynomials that are (polynomial) multiples of the linear polynomial t - a, when a is given. Does the isomorphism class of the quotient space F[t]/U_a depend on a? Let F be a field and S a set. Recall that F^S denote the vector space (over F) of all maps from S to F. Let U denote the subspace F^<S> consisting of all maps from S to F with finite support. How might the quotient vector space F^S/F^<S> be described intuitively to someone who is not familiar with the quotient construction? 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 finite-dimensional 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 non-zero. 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