Math 1600A
Homework 6
Fall 2024
Instructions for GradeScope:
You may handwrite your answers or type them. Only type solutions if you are able to type the appropriate symbols. If typed, use at least a 12pt font, with one question per page. You can then directly submit the pdf file to GradeScope.
If handwritten on a tablet:
● You can save your handwritten solutions to pdf files and upload the pdf files to the GradeScope website.
● You then do not have to scan your solutions. If handwritten on paper:
● Don’t erase or cross out more than a word or two.
● Only write on one side of the page. Use dark, large, neat writing.
● If you have a real scanner, great! Most people will scan their solutions using their phone. Use one of the recommended apps: For iOS, use Scannable by Evernote or Genius Scan. For Android, use Genius Scan. Do not just take regular photos. Read the pages at the end of this document for GradeScope’s instructions.
● When scanning, have good lighting and try to avoid shadows.
● It is best to have one question per scan. If the solution is short, fold the page in half and scan just the half it is on, so there isn’t so much blank space. Or, you can scan and then crop the scan.
You don’t need to scan parts (a), (b), etc. separately or put them on separate pages.
● It works well to scan each question separately and produce one pdf file with one question per page. Most scanning apps will automatically combine your images into one pdf file.
● You must check the quality of your scans afterwards, and rescan if needed.
You must access Gradescope by clicking on the link on the course OWL Brightspace page. You do not need to create a Gradescope account or use a course access code.
You can resubmit your work any number of times until the deadline.
Don’t forget to accurately match questions to pages. If you do this incorrectly, the grader will not see your solution and will give you zero.
See the GradeScope help website for lots of information: https://help.gradescope.com/
Select “Student Center” and then either “Scanning Work on a Mobile Device” or “Submitting an Assign- ment” .
Instructions for writing solutions:
● Homework is graded both on correctness and on presentation/style.
● Do not cross out or erase more than a word or two. If you write each final solution on a new page, it’s easy to start over on a fresh page when you’ve made a large error.
● Show all the steps of your calculations and justify any statements made. However, do not show any rough work that isn’t needed to justify your answers.
● You should do the work on your own. Read the course syllabus for the rules about scholastic of- fences, which include sharing solutions with others, uploading material to a website, viewing material of others or on a website (even if you don’t use it), etc. The penalty for cheating on homework will be a grade of 0 on the homework set as well as a penalty of negative 5% on the overall course grade.
[5] 1. For each of the following statements, determine whether the statement is true or false. If the statement is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false, give an explicit numerical counterexample to the statement. No marks will be given without correct justification.
(a) If A and B are symmetric 3 × 3 matrices with real entries, then 2A − 3B is a symmetric 3 × 3 matrix.
(b) If A is a 5 × 3 matrix with real entries, then the rows of A must be linearly dependent.
(c) If A and B are 2 × 2 matrices with real entries such that A2 = B2 , then A = ±B .
(d) If A is a 2 × 3 real-valued matrix with a zero column and B is any 3 × 3 real-valued matrix, then AB must have a zero column.
(e) If vectors v, w ∈ R3 are non-zero and orthogonal, then {v, w} is linearly independent.
2. A network of irrigation canals is shown in the figure below with water flow measured in litres per second:
[4] (a) Set up and solve a system of linear equations to find the possible flows f1,f2,f3 , f4 and f5 .
[4] (b) Assume all the flows to be non-negative. If the flow f5 is regulated to be 30 L/s, then determine the minimum and maximum flows along each of the other canals.
3. Consider the set of vectors
[5] (a) Describe span(S) both algebraically and geometrically. Show your work.
[2] (b) Find a vector in R3 which is not in span(S). Use (a) to justify your answer.
4. Consider the following matrices:
[2] (a) Find row 2 and column 3 of AB.
[4] (b) Express row 2 of AB as a linear combination of the rows of B . Express column 3 of AB as a linear combination of the columns of A.
[4] 5. Determine whether the following set of real matrices, S = {A1 , A2 , A3 }, where:
is linearly independent or linearly dependent. If it is linearly dependent, find a non-trivial linear combination of the matrices A1 , A2 , A3 which is equal to the zero matrix.