Homework and Syllabus
Linear Algebra
MATH 2130 Fall 2022
Homework must be submitted via Canvas, and is due at the start of class, with your name and homework number on it to receive credit. You will be graded on the clarity of your exposition. Messy, disorganized, or poorly written assignments will not receive credit.
You will receive 1 point of Extra Credit on your homework for doing it in LaTeX (you must upload both the .tex and .pdf files to Canvas to get the extra credit).
Please read
the suggested texts before class, and then after class
make sure to attempt the homework for the sections we
covered that day.
You may find it useful to use a computer
algebra system to check your matrix computations. The
program Mathematica,
for instance, is available free to students via
the University of Colorado.
You may find the Math Academic Resource Center ("MARC") to be useful as a meeting point for discussing homework, and a place to get some further help understanding the material.
An asterix * indicates that a homework assignment has not been finalized.
| Date | Topics |
Reading |
Homework |
| L1 Monday August 22 |
Introduction to the course, and
overview of mathematical notation A brief review of some mathematical notation, including sets, subsets, unions, intersections, products, and maps. Matrix addition and multiplication, and determinant and inverse formulas for small matrices. |
Please see the
references for LaTeX at the bottom of this webpage. You may also want to take a look at The Not So Short Introduction to LaTeX 2e |
|
| L2 Wednesday August 24 |
Linear
equations in linear algebra Systems of linear
equations, row reduction, reduced row echelon form. |
D. Lay, S. Lay, and
J. McDonald, Linear Algebra
and Its Applications (5th Edition),
Pearson, 2016 Section 1.1-2 Here are some slides on the Reduced Row Echelon form of a matrix. |
HW1a Lay Section 1.1: 1, 3, 16. Lay Section 1.2: 1, 7, 29. |
| L3 Friday August 26 |
Linear
equations in linear algebra continued. Vector equations, span, matrix equations, solution sets of linear systems. |
Lay Sections
1.3-5 Here are some slides on solving systems of linear equations with the modified matrix. |
HW 1 DUE You must upload your homework to Canvas in .pdf format. Here is a LaTeX template for your homework, if you want it .pdf, .tex, .bib. Solution to 1.1.16 .pdf Solution to 1.2.7 .pdf HW2a Lay Section 1.3: 2, 22, 33. Lay Section 1.4: 10, 12, 15. |
| L4 Monday August 29 |
Linear
equations in linear algebra continued. Solution sets of linear systems, applications of linear systems (1.6). |
Lay Sections 1.5-6 | HW2b Lay Section 1.5: 2, 7, 20. Lay Section 1.6: 3, 6, 11. |
| L5 Wednesday August 31 |
Linear
equations and linear algebra continued. Linear independence, introduction to linear maps ("transformations"). |
Lay Sections 1.7-8 | HW2c Lay Section 1.7: 2, 6, 22. Lay Section 1.8: 3, 22, 29. |
| L6 Friday September 2 |
Linear
equations and linear algebra continued. The matrix of a linear map ("transformation"). |
Lay Section 1.9 |
HW 2 DUE Solution to 1.3.22 .pdf Solution to 1.7.2 .pdf HW3a Lay Section 1.9: 2, 3, 14, 24, 29, 30. |
| Monday
September 5 |
LABOR DAY |
NO CLASS |
LABOR DAY |
| L7 Wednesday September 7 |
Linear
equations and linear algebra continued. Linear models in business, science, and engineering (1.10). Matrix algebra Matrix operations. |
Lay Sections 1.10 Lay Section 2.1 |
HW3b Lay Section 1.10: 1, 10. Lay Section 2.1: 2, 4, 16, 32. |
| L8 Friday September 9 |
Matrix
algebra The inverse of a square matrix, characterizations of invertible matrices. |
Lay Sections 2.2-3 | HW 3
DUE Solution to 1.9.2 .pdf Solution to 1.9.29 .pdf HW4a Lay Section 2.2: 5, 10, 13. Lay Section 2.3: 4, 12, 27. |
| L9 Monday September 12 |
Matrix algebra continued Partitioned matrices, matrix factorizations (2.5), the Leontief Input-Output model (2.6), applications to computer graphics (2.7). |
Lay Sections 2.4-7 |
HW4b Lay Section 2.4: 2, 5. Lay Section 2.5: 1, 24. Lay Section 2.6: 4. Lay Section 2.7: 5. |
| L10 Wednesday September 14 |
Matrix algebra continued Subspaces of real n-space, kernel ("null space"), dimension, and rank. |
Lay Sections 2.8-9 | HW4c Lay Section 2.8: 2, 5, 16. Lay Section 2.9: 4, 9, 18. |
| L11 Friday September 16 |
Determinants Introduction to determinants, properties of determinants, minor, cofactor, cofactor matrix (transpose of the "adjutant" or "classical adjoint") volume and linear transformations (3.3). |
Lay Sections 3.1-3 | HW 4 DUE Solution to 2.3.27 .pdf Solution to 2.9.9 .pdf HW5a Lay Section 3.1: 2, 11, 15. Lay Section 3.2: 2, 7, 28. |
| L12 Monday September 19 |
Review |
|
|
| L13 Wednesday September 21 |
Review
practice exam |
Practice exam and solutions. |
|
| Friday September 23 |
MIDTERM I | MIDTERM I |
|
| L14 Monday September 26 |
Vector
spaces Vector spaces and subspaces, linear maps ("transformations"), kernel ("null space"), image. |
Lay Sections 4.1-2 You may also want to take a look at this .pdf |
HW5b Lay Section 4.1: 2, 6, 24. Lay Section 4.2: 2, 8, 26. |
| L15 Wednesday September 28 |
Review
exam |
||
| L16 Friday September 30 |
Vector
spaces continued Linearly independent sets, bases, coordinate systems. Dimension. |
Lay Sections 4.3-5 | HW 5 DUE Solution to 4.1.6 .pdf Solution to 4.2.26 .pdf HW6a Lay Section 4.3: 4, 15. Lay Section 4.4: 2, 10. Lay Section 4.5: 1, 9. |
| L17
Monday October 3 |
Vector spaces continued Dimension, rank, change of basis. |
Lay Sections 4.5-7 | HW6b Lay Section 4.6: 2, 13, 18. Lay Section 4.7: 1, 6, 11. |
| L18 Wednesday October 5 |
Vector
spaces continued Applications to difference equations (4.8), and Markov chains (4.9). |
Lay Sections 4.8-9 | HW6c Lay Section 4.8: 2, 5, 19. Lay Section 4.9: 4, 10, 18. |
| L19 Friday October 7 |
Eigenvalues
and eigenvectors Eigenvalues and eigenvectors, the characteristic polynomial, diagonalization. |
Lay Sections 5.1-3 | HW 6 DUE Solution to 4.7.6 .pdf Solution to 4.9.18 .pdf HW7a Lay Section 5.1: 2, 7, 22. Lay Section 5.2: 1, 9, 22. |
| L20 Monday October 10 |
Eigenvalues
and eigenvectors continued Diagonalization. |
Lay Section 5.3 | HW7b Lay Section 5.3: 1, 5, 22, 24, 27, 31. |
| L21 Wednesday October 12 |
Eigenvalues
and eigenvectors continued Eigenvectors and linear maps ("transformations"). |
Lay Sections 5.4 | HW7c Lay Section 5.4: 2, 8, 9, 12, 14, 25. |
| L22 Friday October 14 |
Eigenvalues
and eigenvectors continued Complex eigenvalues. |
Lay Sections 5.5 | HW 7 DUE Solution to 5.3.27 .pdf Solution to 5.4.25 .pdf HW8a Lay Section 5.5: 1,2, 8, 12, 23, 24 (Exercise 24 is false as stated -- see the solution below). |
| L23 Monday October 17 | Eigenvalues
and eigenvectors continued Discrete dynamical systems (5.6), applications to differential equations (5.7). |
Lay Sections 5.6-7 | HW8b Lay Section 5.6: 1, 5, 12. Lay Section 5.7: 1, 4, 9. |
| L24 Wednesday October 19 | Orthogonality
and least squares Inner product, length, and orthogonality. |
Lay Section 6.1 | HW8c Lay Section 6.1: 2, 10, 14, 20, 22, 30. |
| L25 Friday October 21 |
Orthogonality
and least squares continued Orthogonal sets, orthogonal projections. |
Lay Sections 6.2-3 | HW 8 DUE Solution to 5.5.23 .pdf Solution to 5.5.24 .pdf HW9a Lay Section 6.2: 3, 11, 26. Lay Section 6.3: 1, 12, 22. |
| L26 Monday October 24 |
Review |
||
| L27 Wednesday October 26 | Review practice
exam |
Practice exam and
solutions. |
|
| Friday
October 28 |
MIDTERM II | MIDTERM II |
|
| L28
Monday October 31 |
Orthogonality
and least squares continued Gram--Schmidt. |
Lay Section 6.4 | HW9b Lay Section 6.4: 2, 4, 10, 17, 18, 19. |
| L29
Wednesday November 2 |
Review
exam |
||
| L30 Friday November 4 |
Orthogonality and
least squares continued Least squares problems. |
Lay Section 6.5 | HW 9 DUE Solution to 6.2.26 .pdf Solution to 6.4.19 .pdf HW10a Lay Section 6.5: 2, 6, 13, 18, 19, 20. |
| L31 Monday November 7 |
Orthogonality
and least squares continued Applications to linear models (6.6). |
Lay Section 6.6 | HW10b Lay Section 6.6: 1, 2, 5, 7a, 14, 19. |
| L32 Wednesday November 9 |
Orthogonality
and least squares continued Inner product spaces |
Lay Section 6.7 | HW10c Lay Section 6.7: 1, 3, 7, 9, 20, 21. |
| L33 Friday November 11 |
Orthogonality
and least squares continued Applications of inner product spaces |
Lay Section 6.8 | HW 10 DUE Solution to 6.6.14 .pdf Solution to 6.6.19 .pdf HW11a Lay Section 6.8: 1, 2, 5, 6, 7, 13. |
| L34 Monday November 14 | Symmetric
matrices and quadratic forms Diagonalization of symmetric matrices |
Lay Section 7.1 | HW11b Lay Section 7.1: 2, 8, 14, 24, 25, 27. |
| L35
Wednesday November 16 |
Symmetric matrices and
quadratic forms continued Quadratic forms |
Lay Section 7.2 | HW11c Lay Section 7.2: 1, 3, 5, 7, 23, 24. |
| L36 Friday November 18 |
Review
homework |
HW 11 DUE Solution to 7.2.23 .pdf Solution to 7.2.24 .pdf |
|
| November 21--25 | THANKSGIVING BREAK |
NO CLASS |
THANKSGIVING BREAK |
| L37 Monday November 28 |
Symmetric
matrices and quadratic forms continued Constrained optimization |
Lay Section 7.3 | HW12a Lay Section 7.3: 2, 4, 5, 9, 12, 13. |
| L38 Wednesday November 30 |
Symmetric
matrices and quadratic forms continued Singular Value Decomposition |
Lay Section 7.4 | HW12b Lay Section 7.4: 1, 2, 4, 8, 17, 18. |
| L39 Friday December 2 |
Symmetric
matrices and quadratic forms continued Applications to image processing and statistics |
Lay Section 7.5 | HW 12 DUE Solution to 7.3.13 .pdf Solution to 7.4.18 .pdf HW13a Lay Section 7.5: 1, 2, 3, 4, 9 13. |
| L40 Monday December 5 |
Review |
||
| L41 Wednesday December 7 | Review exam |
Practice exam and solutions. |
HW13 DUE (note the unusual
date) Solution to 7.5.3 .pdf |
| Friday December 9 | READING DAY |
NO CLASS |
NO CLASS |
| Sunday
December 11 |
FINAL EXAM 7:30
PM -- 10:00 PM CLUB 4 (Lecture Room) |
FINAL EXAM |
I strongly encourage
everyone to use LaTeX for typing homework. If you have
a mac,
one possible easy way to get started is with texshop.
If you are using linux,
there are a number of other possible ways to go, using
emacs, ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something online.
Here is a sample homework file to use: (the .tex
file, the .bib
file, and the .pdf
file). This site can help you find LaTeX symbols
by drawing: http://detexify.kirelabs.org/classify.html.
You may also want to try https://cocalc.com
(formerly https://cloud.sagemath.com/)
or https://www.overleaf.com
(formerly https://www.sharelatex.com/)
for a cloud version.