Tentative Problem sheet - Math 6332 - Advanced Applications in Linear

Algebra

Module 3 - Vector Space Introduction

Module 3 Primary topics: Subspaces of Vector spaces, Linear Independence, Span, and Basis;

Module 3 Applications: Basis for Polynomial Spaces, Designing a Ski Slope, Making Movies

Assignments: Due Saturday, June 24, midnight (approximately 11:59 pm)

General guidelines. Show work for by hand problems. When implementing Matlab and rref write

down the matrix and result from matlab, what it means is your part. Implement Matlab and rref

to sort out linear independence and bases of vector sets. Turn in indicated portions of the Lab

Exercises with the Matlab results filled in in the provided areas, provide Matlab output, graphs,

when requested.

Note: provide bases and independent or dependent sets as "sets in vector form", the original

vectors or vectors from the actual vector space, not the vectors created to allow use of Matlab.

1. Shores: 3.2 - 1,5, 6, 7, 11, 14(a,b)

2. Shores: 3.3- 1(b,c), 5(a,b), 6(a,b), 7, 8, 11, 12

3. Shores: 3.5 - 1, 3, 12, 17

4.

Shores: 3.6 – 1, 2, 5, 6(a,b), 13 (use Matlab; for examples see the Lab 5 and Lab 9

examples portions)

Additional problems:

Matlab lab Assignments: Matrix Transformations (the lab examples and exercises files are in the

Matlab materials folder.)

5. Lab 16-Matrix Transformations. Linear Algebra and Movies (Animation.)

Exercises 2, 4, 5, 7

Matlab lab Assignments: (the lab example and exercise files are in the Matlab materials folder.)

6. Lab 9 Basis. Exercises 1, 2, 4(a, d, e)

7. Explain the connections between span of a set of vectors, a basis for the span of the

set, and the dimension of the space spanned by the set./n"Ski Jump Design"

8. Example of how to find polynomial and plot it with Matlab details to help once you

get your points for the Ski Jump Design Problem:

You have been ordered to find a curve through the points (1,7), (1.5,6), and (2.4,9).

Since there are three points the polynomial of least order that fits all data will be

quadratic. This leads to the following system of equations, solved using rref on the

augmented matrix. The resulting polynomial is plotted in Matlab, see the commands

below to evaluate then plot P(x).

P(x) = ax² +bx+c

P(1) = 7; P(1.5) = 6; P(2.4) = 9; so

la + lb + lc = 7

2.25a +1.5b+c=6

5.76a+2.4b+c=9

⇒a=80/21, b = -242/21, c=103/7

The solution is determined in Matlab using the rref command as with A the

coefficient matrix and B the right hand side vector as:

aug = [A B]; format rat; rref(aug)

The following commands create a plot of the interpolating polynomial in Matlab. The

original data values are plotted with an asterix to verify the polynomial does its job.

datax [1 1.5 2.4];datay=[769];

coefs=[80/21-242/21 103/7];

x=0:.125:3;y=coefs(1)*x.*x+coefs(2)*x+coefs(3); % x^2 is x.*x not x*x

plot(x,y,'b', datax,datay,'m*');

title('Prof. D"s polynomial, blue; given data, magenta *');grid;shg

15

14

13

12

11

10

9

8

7

6

5

0

Prof. D's polynomial, blue, through data, magenta

0.5

1

1.5

2

2.5

3/nThe Ski Jump Problem:

Now for the ski problem details. You have been hired to design a ski jump. You will

determine a polynomial of the smallest degree with the following conditions. The ski

jump starts at a height of 33.5 m and finishes at a height of 3.2 m. From start to finish,

the ski jump covers a horizontal distance of 40 m. A skier at the top of the jump will

start off horizontally and will take off at the end at an angle of 30 degrees from the

horizontal. Find and plot the polynomial whose graph is a side view of the ski jump.

Be careful, points are not all known, but you do know slope information, recall your

calculus and basic triangle trigonometry. Determine the coefficients of the

polynomial, write out the polynomial and plot the polynomial in Matlab over the

domain [0, 40]. Provide a copy of your ski jump plot.

Additional comments on homework

Given that the focus of this course is of the applications associated with linear algebra

methods, the problems that need be submitted for assessment purposes (grading) are

first of all the application problems. Secondarily, some, if not all, of the book's

problems indicated will be graded, they are largely for you to work, practice and

review concepts, you will submitted them as well for assessment purposes. Do be

aware that if you do not work through each of the selected book problems you may

not be sufficiently refreshed on material you should have seen in your undergraduate

mathematics courses, especially linear algebra topics. Problems have been selected to

review those procedures and theories helpful in completion and comprehension of the

applications of the module.

For Labs 16 and 9, write out your responses in the blanks and spaces of the exercises

portions of the labs. It is likely best to print out the exercises pages and write

responses in the blanks or space provided below or adjacent to the problem, it is not

necessary nor desired to include the Matlab commands and output. The commands

are those of the examples portion of the labs, results are the goal. Provide plots only

when/where requested. Do remember that Matlab does not put brackets [] around

arrays but you must do so when you write them down in the response areas. Sets of

vectors should be in set notation, e.g., {A1,A3,A4}, see the examples documents for

examples of expected presentation of responses. Final reminder, when writing down

sets of vectors, put the vectors in their actual form, not the vectors used in Matlab as

the trick to see who is linearly independent, thus who belongs in a basis. See example

2 of Lab 9 examples.

Reminder on submission of work

The nature of your homework, your handwriting and how fast you type will likely

impact how you choose to "write up" your homework. As is the case in mathematics,

it is often easier to just work some of the problems by hand with paper and a pen or

pencil. This is fine as long as it is legible. It may be helpful to check your work with/ntechnology, however you must show your work unless use of Matlab is indicated or it

is a Matlab problem/project. If a homework problem is such that Matlab was used,

you will likely want to cut and paste your output into a word document and type up

your resulting answer. That is also fine and submissions may be a combination of the

two. Please do not paste in pages and pages of preliminary output or lots of large

arrays, it is sufficient to indicate commands implemented and results. Regardless of

what your final work looks like, we need it saved as a single PDF document to be

uploaded into the drop box. Access to a scanner will likely be helpful in this process

if your work is a combination of Word and handwritten work. Please remember to

save your homework as a single pdf file so that you upload only one

document. Whether your submission is printed and graded by hand or graded on an

Ipad Pro either is much easier to deal with if a PDF document is submitted.

What is to be posted up on WTclass for assessment purposes:

Shores: 3.2-6, 14(a,b); 3.3 – 1(b,c), 6(a,b), 8, 12; 3.5-3, 12; 3.6-2, 6(a,b);

Additional Problems: 5, 6, 8