Date Assigned
|
Date Due
|
Section
(Page)
|
Assignment
|
Comments
|
| 17 Jan |
26 Jan |
§1.1
(p. 17) |
PP: # 1b, 2c, 3c, 6b, 7, 8, 13, 15, 17, 19a-e
|
-
This section introduces the language of linear systems and develops
techniques used in solving them. Gaussian elimination is explained
initially for 3x3 linear systems. Once the logic of the algorithm
has been established, we shift to using row operations on the augmented
matrices of arbitrary linear systems.
-
These problems are all pencil and paper problems.
-
I encourage you to use Maple if you want help visualizing the problem.
-
Solution Key:
PP: [PDF]
|
| 19 Jan |
26 Jan |
§1.2
(p. 25) |
MM: # 2, 3, 5
|
-
This section looks at the geometry of linear systems in two and
three unknowns. The emphasis here is on working back and forth
between the algebraic representations of the system and the
corresponding geometry of lines and planes.
-
New VLA commands: Drawlines, Drawplanes,
and Plotpoints.
-
These are Maple problems. Remember that the problems are included at
the end of the tutorial worksheet, and as a separate worksheet. It
can be sufficient to simply print out the worksheet.
-
You may need to do some manual calculations,
but some part of each problem is designed to be solved with the help
of Maple.
-
Solution Key:
MM: [Worksheet]
|
| 22 Jan |
26 Jan |
§1.3A
(p. 34) |
MM: # 1, 3, 4, 5
|
-
This section introduces several Maple commands for solving linear
systems.
-
New VLA commands: Rowop, Pivots, Reduce,
and Backsolve.
-
We will talk about # 2 and # 6 in class.
-
Modified worksheet, showing the use
of labels - and the solutions to #2 and #6. Remember to use ctrl-l to enter a label. Then, a right-mouse
click on the label will allow you to toggle between the label and the
value associated with the label.
-
Solution Key:
MM: [Worksheet]
|
| 23 Jan |
30 Jan |
§1.4
(p. 54) |
MM: # 2
|
-
This section provides a sampling of applications that are particularly
accessible and easy for students to appreciate. The applications include
curve fitting by a single polynomial and by a cubic spline, and
discretization of a steady-state temperature distribution in a planar
region.
-
New VLA commands: Genmatrix and Drawspline.
-
While only #2 is due next week, you should look at #3 and #5.
These have been moved to the lab assignment for the following week.
-
Solution Key:
MM: [Worksheet]
|
| 24 Jan |
2 Feb |
§2.1
(p. 69) |
MM: # 2, 3ab, 4, 5
PP: # 5, 8, 9, 11
|
-
This section reviews the geometry of vectors in the plane and
in 3-space. Vector addition, scalar multiplication, translation
by vectors, and teh vector form of lines are discussed and
illustrated.
-
New VLA commands: Drawvec, Drawmatrix,
Vectranslate, and Vectorline.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 26 Jan |
2 Feb |
§2.2
(p. 81) |
MM: # 2, 3, 4ab
PP: # 1b, 3, 4a, 8, 10
|
-
This section introduces the concepts of linear combination and span
of a set of vectors in and examines these
concepts from both algebraic and geometric perspectives. Also, the
rules of algebra for vectors in are introduced.
-
New VLA commands: Vectorgrid, Gridgame,
and Unitspan.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 29 Jan |
2 Feb |
§2.3
(p. 90) |
MM: # 1, 4, 6
PP: # 4a, 5ab, 8b
|
-
This section is devoted to the interplay of the algebra of
decomposing a solution set in the form
|
| 30 Jan |
6 Feb |
§1.4
(p. 54) |
MM: # 3, 5, 6
Extra Credit: # 7
|
-
This section provides a sampling of applications that are particularly
accessible and easy for students to appreciate. The applications include
curve fitting by a single polynomial and by a cubic spline, and
discretization of a steady-state temperature distribution in a planar
region.
-
Be sure you come to class and learn how to use Maple's D
command to define the derivative of a function. (This will be
easier than using diff.
-
#3 and #5 were moved from last week's lab to this week's lab.
-
Complete #7 for Extra Credit. Read the text to learn how to setup the
necessary equations. The rest should be straightforward.
-
Solution Key:
MM: [Worksheet]
|
| 31 Jan |
5 Feb (class) |
§2.4
(p. 104) |
MM: # 2, 3, 4, 7
PP: # 2a-c, 3a-c, 5abd, 8
|
-
This section introduces the fundamental concept of linear
independence of a set of vectors. The concept is introduced
geometrically in first, later in
the section, the complete, algebraic definition is presented
for vectors in .
-
New VLA commands: none
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 30 Jan |
5 Feb (class) |
§2.5
(p. 111) |
PP: # 2, 4, 5, 6c, 7
|
-
This section summarizes and reviews the important concepts,
definitions, and methods from teh first four sections of the
chapter. The second half of the section contains three theorems
on span and linear dependence; these provide good practice using
the definitions and are used in Chapter 5 in the proofs of
theorems on basis and dimension.
-
New VLA commands: none
-
Solution Key:
PP: [PDF]
|
| 6 Feb |
7 Feb |
Lab 3 |
Worksheet
|
- Follow the directions in the worksheet.
- This is the modified worksheet. The matrix M7 has been
changed (so that it is not the same as M6).
This is the only change to the file.
|
| 7 Feb |
7 Feb |
Exam 1 |
all of chapters 1 and 2
|
-
Solution Key:
[PDF]
-
Supplemental Sheet:
[PDF]
|
| 9 Feb |
16 Feb |
§3.1
(p. 123) |
MM: # 2, 4, 5, 6
PP: # 2, 3, 5, 7(b), 9, 10
|
-
This section introduces the matrix-vector product and explores its
connctions with linear combinations of vectors, span, linear
independence, linear systems, and the decomposition of the solution
set of a linear system discussed in Section 2.3.
-
New VLA commands: Randmat
- These problems should not require a lot of computation.
- Pay careful attention to the specific questions that are asked,
and to the terminology.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 12 Feb |
16 Feb |
§3.2
(p. 136) |
MM: # 1, 5, 7
Extra Credit: # 6
PP: # 3, 7, 10, 16
|
-
This section defines and explores the properties of the product of
two matrices and powers of a square matrix.
-
New VLA commands: Backidmat, Diagmat, Idmat,
Jordanmat, LetterL
- A few of these problems ask you to do some experiments, and
to make some observations based on these experiments. You do
not have to show all of your experimental data, but should include
enough to support your conclusions.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 13 Feb |
20 Feb |
§3.4
(p. 158) |
MM: # 2, 3
PP: # 3
|
-
Topics covered include stochastic and regular stochastic matrices,
steady-state vectors, and limit vectors and limit matrices. This
topic is revisited and extended in Section 7.3, Discrete Dynamical
Systems.
- We will spend two weeks on Markov Chains.
- There is some new terminology to learn, including
probability vector and regular stocastic matrix.
|
| 14 Feb |
23 Feb |
§3.3
(p. 146) |
PP: # 1, 2, 3(a), 5
|
-
In this section, the rules of matrix algebra, including those for
the transpose of a matrix, are stated and applied. Proofs for many
of the rules are included, easier proofs are left to the problems.
- When you have to come up with an example it is often helpful to
keep in mind the meaning of matrix multiplication in terms of
linear combinations.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 16 Feb |
23 Feb |
§3.5
(p. 171) |
MM: # 1, 3, 7
PP: # 1(a), 3(a), 4, 8, 10, 13
|
-
This section introduces the matrix inverse and explores many of its
properties. The Invertibility Theorem describes the relationship
between invertibility of a matrix and the fundamental vector concepts
of linear independence and span.
-
New VLA commands: Inverse
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 20 Feb |
27 Feb |
§3.4
(p. 158) |
MM: # 5, 6
PP: # 4
|
- This is the second week working with Markov Chains.
-
Topics covered include stochastic and regular stochastic matrices,
steady-state vectors, and limit vectors and limit matrices. This
topic is revisited and extended in Section 7.3, Discrete Dynamical
Systems.
|
| 21 Feb |
2 Mar |
§3.6
(p. 180) |
PP: # 1(a), 2, 3
|
-
The section reviews the theory of matrix inverses developed in
Section 3.5 and provides all the remaining proofs.
- For #1(a), what property must a matrix and its inverse satisfy?
Show that the two given matrices satisfy this property (using the
additional assumption that ).
-
Solution Key:
PP: [PDF]
|
| 23 Feb |
2 Mar |
§4.1
(p. 198) |
PP: # 2, 4, 5, 6
|
-
In this section, matrix transformation from
to are introduced. Geometric transformations
of and are
emphasized in the examples. Kernal and range of a matrix transformation
are also defined; again, the emphasis is on geometric visualization.
-
Solution Key:
PP: [PDF]
|
| 26 Feb |
2 Mar |
§4.2
(p. 216) |
MM: # 1, 2(a), 5
Extra Credit: MM # 3, 4
PP: # 1, 2, 3, 5, 6, 8, 10, 11(a)-(c)
|
-
This section introduces five types of geometric transformations of the
plane: scalings, shears, rotations, reflections, and projections.
Composite transformations are presented algebraically and geometrically.
Animations are used to illustrate transformations dynamically.
-
New VLA commands: Clock, Movie, Projectmat,
Reflectmat, Rotatemat, Transform,
Xshearmat, Yshearmat
- This is a long section. You need to try to pick out the fundamental
facts and be able to use this knowledge to build additional facts.
I will be trying to help you with this, but you need to be working
on it also.
- Some of the Pencil and Paper exercises build upon the results you
will find in the Maple/Mathematica exercises
(e.g., PP #2 is related to MM #2).
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 27 Feb |
6 Mar |
§4.5
(p. 253) |
MM: # 1, 2(a)-(d)
|
-
Homogeneous coordinates are introduced as a technical device to
incorporate translations in the framework of matrix transformations;
no projective geometry is used.
- Linear algebra is very important in computer graphics. This week,
and next, you will learn some of the basics how linear algebra
is used to create animations.
- Homogeneous coordinates are introduced as a way to be able to
handle translations with matrix multiplication.
- Look at each step in the animation. Figure out what transformation
was applied, then determine the appropriate matrix that corresponds
to that transformation (in homogeneous coordinates).
-
Solution Key:
MM: [Worksheet]
|
| 28 Feb |
5 Mar
(class) |
§4.3
(p. 230) |
MM: # 1, 3, 4, 5, 6, 7
PP: # 1(a)-(c), 3
|
-
This section extends the discussion of geometric transformations
to 3-space. Kernel and range are explored further.
-
The method introduced in MM #5 can be helpful with the solution
of MM #3(c).
-
New VLA commands: Projectmat3d, Reflectmat3d,
Rotatemat3d
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 2 Mar |
5 Mar
(class) |
§4.4
(p. 236) |
PP: # 1, 2, 3, 5, 7
|
-
In this section, the definition of a linear transformation from
to and the
definition of the algebraic operations on those transformations
are presented. Also included are proofs of the correspondence
between operations on linear transformation and operations on
matrices.
- This is a long section. You need to try to pick out the fundamental
facts and be able to use this knowledge to build additional facts.
I will be trying to help you with this, but you need to be working
on it also.
-
Solution Key:
PP: [PDF]
|
| 6 Mar |
20 Mar |
§4.5
(p. 253) |
MM: # 3, 5
Extra Credit: # 6
|
-
Homogeneous coordinates are introduced as a technical device to
incorporate translations in the framework of matrix transformations;
no projective geometry is used.
- These problems are not difficult --- provided you break them down
into the appropriate steps.
- Use the examples in the text as a guide!
- Do not be afraid to experiment.
- For the Extra Credit, save your work to a Maple worksheet and
e-mail it to me.
|
| 7 Mar |
7 Mar |
Exam 2 |
all of chapters 3 and 4
|
|
| 9 Mar |
17 Mar |
Exam 2 Retest |
all of chapters 3 and 4
|
- Retake Exam 2 and turn in your solutions no later than the
beginning of class on Monday, March 17.
- You can use your book and the computer, but do NOT talk with
any other student, professor, or other animate object.
|
| 9 Mar |
23 Mar |
§5.1
(p. 261) |
PP: # 1, 2, 7, 8, 9, 15, 20, 22
|
-
Subspaces of are introduced in this
section. Examples include the span of a set of vectors, the kernel
and range of a matrix transformation, and subspaces of
.
-
Solution Key:
PP: [PDF]
|
| 19 Mar |
23 Mar |
§5.2
(p. 268) |
MM: # 1, 2, 4, 5
PP: # 1, 4, 6, 10, 11, 13, 15, 18
|
-
Basis and dimension are disucussed in this section. Several geometric
examples are considered, and a general method is presented for finding
a basis of a null space by decomposition of a solution set. Also,
coordinates of a vector relative to a basis are introduced.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 20 Mar |
27 Mar |
§5.6
(p. 295) |
MM # 1, 2
|
-
This section demonstrates how bases of subspaces can be used to
analyze the flow through a network. The two examples considered
are a traffic network of streets and a network of oil pipelines.
-
The investigation of this topic will continue next week.
-
Solution Key:
MM: [Worksheet]
|
| 21 Mar |
30 Mar |
§5.3
(p. 274) |
PP: # 1, 3, 5, 7
|
-
This section presents proofs of Theorems 3 through 6 (about
span, basis, and dimension).
-
Solution Key:
PP: [PDF]
|
| 26 Mar |
30 Mar |
§5.4
(p. 283) |
MM: # 1, 2, 4, 5
PP: # 3, 4, 7, 8, 9
|
-
In this section we introduce the column space, row space, and null
space of a matrix and provide students with extensive practice finding
bases of these subspaces by a variety of methods.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 27 Mar |
3 Apr |
§5.6
(p. 295) |
MM # 4, 6
|
-
This section demonstrates how bases of subspaces can be used to
analyze the flow through a network. The two examples considered
are a traffic network of streets and a network of oil pipelines.
-
This week's lab is a continuation from last week.
|
| 28 Mar |
6 Apr |
§5.5
(p. 289) |
PP: # 1, 4, 5, 6, 8
|
-
This section reviews the concepts, methods, and facts developed in
Section 5.4. Theorems 7 and 8 are the fundamental theorems describing
the relationships among the dimensions of the column space, row space,
and null space.
-
Solution Key:
PP: [PDF]
|
| 30 Mar |
6 Apr |
§5.7
(p. 304) |
PP: # 1, 4, 7, 11, 16(acd), 17, 18
|
-
In this section, abstract vector spaces and their properties are
presented. The main examples discussed are the vector space of
by matrices, the vector space of
real-valued functions defined on an interval, and subspaces of
these spaces.
-
Solution Key:
PP: [PDF]
|
| 2 Apr |
6 Apr |
§6.1
(p. 311) |
PP: # 2(abcd), 4, 5(abcd), 7
|
-
This short section introduces the determinant of an by
matrix via matrix cofactor expansions.
-
Solution Key:
PP: [PDF]
|
| 2 Apr |
10 Apr |
§6.2
(p. 320) |
MM: # 1, 3, 4
PP: # 2, 4, 5, 7
|
-
This section explores the usual elementary properties of the determinant
--- the effect of row operations, the criterion for invertibility, and
the algebraic properties. The geometric meaning of the determinant of
2 by 2 and 3 by 3 matrices is also investigated.
-
Solution Key:
PP: [PDF]
|
| 6 Apr |
10 Apr |
§6.3
(p. 327) |
PP: # 4
|
-
This section reviews the introductory material in Section 6.1 and
gives proofs of Theorems 3 through 6 from Section 6.2.
-
Solution Key:
PP: [PDF]
|
| 9 Apr |
20 Apr |
§7.1
(p. 339) |
MM: # 1, 5, 6 (Extra Credit: #2)
PP: # 1, 3, 6, 7, 8, 9, 17
|
-
This section introduces eigenvalues, eigenvectors, and eigenspaces and
explores these concepts algebraically and geometrically.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 11 Apr |
20 Apr |
§7.2
(p. 352) |
MM: # 1, 2, 5a, 6 (Extra Credit: #4)
PP: # 1, 2, 3, 5, 7, 9, 12 (Extra Credit: #8)
|
-
This section introduces the characteristic polynomial. It also includes
several explorations in which patterns of eigenvalues and eigenvectors
are sought.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 13 Apr |
13 Apr |
Exam 3 |
all of chapters 5 and 6
|
|
|
| 16 Apr |
27 Apr |
§7.4
(p. 386) |
MM: # 2, 3, 4 (Extra Credit: #8)
PP: # 1, 2, 5, 6, 8a
|
-
In this section, diagonalization and similar matrices are introduced.
The geometric meaning of the composite transformation is
thoroughly explored.
-
Solution Key:
MM: [Worksheet]
PP: [PDF]
|
| 17 Apr |
24 Apr |
§7.3
(p. 320) |
MM # 1
|
-
This section uses eigenvector bases to analyze the long-term
behavior of solutions to discrete dynamical systems.
-
This week's lab will be continued next week.
|
| 18 Apr |
27 Apr |
§7.5
(p. 394) |
PP: # 1, 3, 4
|
-
This section reviews the main concepts, methods, and facts developed
in the first four sections of the chapter. The proofs of the harder
theorems are provided here.
-
Solution Key:
PP: [PDF]
|
| 24 Apr |
27 Apr |
§7.3
(p. 320) |
MM # 2, 3
PP # 1, 2, 4 (Extra Credit: #6)
|
-
This section uses eigenvector bases to analyze the long-term
behavior of solutions to discrete dynamical systems.
-
This is the last lab of the semester.
|
| 23 Apr |
30 Apr (Optional) |
§7.7
(p. 411) |
PP: # 7, 9
|
-
This section reviews the main concepts, methods, and facts developed
in the first four sections of the chapter. The proofs of the harder
theorems are provided here.
-
These problems will be counted as Extra Credit.
|
| 25 Apr |
30 Apr (Optional) |
§7.8
(p. 428) |
MM: # 3, 4, 5
PP: # 4, 5, 7
|
-
This section reviews the main concepts, methods, and facts developed
in the first four sections of the chapter. The proofs of the harder
theorems are provided here.
-
These problems will be counted as Extra Credit.
|
| 2 May |
2 May |
Final
Exam |
Chapters 1 - 7, with an emphasis on Chapter 7
|
-
Supplemental Maple Worksheet
-
The final exam is comprehensive, but will place an emphasis on the
most recent material: vector spaces and eigenvalues.
-
As you study, see how the topics interact. Notice how the ideas of
free variables and pivot elements enter into the different discussions
about independence, basis, rank, deficiency, similarity, ....
-
The best way to prepare for the final is to study old homework
assignments.
-
Review Sheet: [PDF]
|