LINEAR ALGEBRA, MATH 122
Reminder: You MUST show your work to get credit.
Project 4:
Linear and nonlinear transformations in R
2
Goal
Illustrate a connection between transformations of geometric shapes in the plane and matrices.
General requirements
You may work alone or with one other person. If you work with someone else, hand in one answer sheet
with both of your names on it.
No groups bigger than two. No collaboration between groups. Please read “My policies on Projects”
posted on the course website.
Write your answers on the answer sheet provided in the last few pages of this document. Staple1
all the
paper showing your neatly presented 2 work to the answer sheet.
Introduction
Using computer graphics has become part of everyday life for many people. One of the mathematical foundations
of computer graphics is the fact that to change a geometric shape on the screen (i.e., in R2
), one needs to apply
some transformation to it. While there are infinitely many different transformations, some of them share one
underlying mathematical fact: they can be implemented by multiplying some matrix by the collection of vectors
representing the geometric shape in question. Thus, any 2 × 2 matrix represents some transformation in the plane.
As you have learned, all such transformations are called linear transformations. Their hallmark is that they
transform straight line segments into (different) straight line segments3
. The reason why this occurs is that multiplication
by a matrix changes — i.e., rotates, reflects, or stretches/shrinks — all vectors in the same manner.
In this Project, you will first create a geometric shape and then will apply transformations to it that take straight
lines into straight lines. You will explore how a linear transformation can be implemented in a computer in two
different ways.
In one of the Bonus problem, you will be given a chance to explore truly nonlinear transformations, which take
straight lines into curved ones. This occurs because, unlike linear transformations, truly nonlinear ones change
different vectors (i.e. line segments) by different rules.
This project makes heavy use of Matlab. If you have any questions about the required Matlab commands, you
should use Matlab resources posted on the course website, especially the first of them, as well as Appendix A in
the textbook. You can also look up the meaning and usage of any command by typing in the command window
help commandname; for example: help repmat.
NOTE: All codes found in this Project description are posted online so that you could use them as templates
for your own codes. See instructions on the Answer sheet about naming your codes.
1Your grade will be reduced by 5% if you hand in a pile of non-stapled sheets.
2
I will reduce your grade by an amount left to my discretion in each particular case if your work is presented in a messy way and I have
to waste time deciphering it.
3The converse of this is not true: as you saw in Section 3.7 and will see in this Project, there are transformations taking straight lines
into straight lines, and yet those transformations are not linear according to the definition given by Section 3.7. There is a trick, mentioned
later in this Project, which allows one to treat those transformations as linear, but we will not consider this in any depth here.
1
Step-by-step instructions:
Part 1 Use MATLAB to draw your first or last name’s initial. The sample code shown below4 produces such a
drawing of the last letter of the Russian alphabet. The mathematical concept used in this code is the fact that
equation x = a + (b ? a)t, where 0 ≤ t ≤ 1, defines a segment [a, b].
% Draw the letter
t=[0:0.01:1];
% parameter used to construct all segments
xmin_1=0.3; ymin_1=0.05; xmax_1=0.7; ymax_1=0.5;
% coordinates of end points of the 1st segment
x_1=xmin_1+(xmax_1-xmin_1)*t; y_1=ymin_1+(ymax_1-ymin_1)*t;
% 1st segment of the letter
xmin_2=0.7; ymin_2=0.05; xmax_2=0.7; ymax_2=0.95;
x_2=xmin_2+(xmax_2-xmin_2)*t; y_2=ymin_2+(ymax_2-ymin_2)*t;
% 2n segment of the letter
xmin_3=0.7; ymin_3=0.95; xmax_3=0.45; ymax_3=0.8;
x_3=xmin_3+(xmax_3-xmin_3)*t; y_3=ymin_3+(ymax_3-ymin_3)*t;
% 3rd segment of the letter
xmin_4=0.45; ymin_4=0.8; xmax_4=0.45; ymax_4=0.65;
x_4=xmin_4+(xmax_4-xmin_4)*t; y_4=ymin_4+(ymax_4-ymin_4)*t;
% 4th segment of the letter
xmin_5=0.45; ymin_5=0.65; xmax_5=0.67; ymax_5=0.47;
x_5=xmin_5+(xmax_5-xmin_5)*t; y_5=ymin_5+(ymax_5-ymin_5)*t;
% 5th segment of the letter
x_letter=[x_1 x_2 x_3 x_4 x_5]; y_letter=[y_1 y_2 y_3 y_4 y_5];
% put all segments together into a letter
figure(401); % open a new figure and name it "figure 401"
plot(x_letter,y_letter,’k’)
axis([0 1 0 1])
% plot the letter inside the rectangle [0 1] x [0 1];
% the option ’k’ specifies the color (black)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
segment
2
segment 3
segment 4
segment 5
Your task: Write a code to draw your initial, and produce a plot of the letter. See the answer sheet for the
format of submission of your work.
4Note that each clarifying comment is added after the line that it clarifies.
2
To help you do Exercises 2 and 3, I will first present an example. You will need to mimic and extend it to do
the remainder of this Project.
Example Let us first shift the initial you have created by 0.8 units horizontally and by ?0.3 units vertically.
This means that we need to add 0.8 to the x-coordinate of each point of the letter and also add ?0.3 to (i.e.,
subtract 0.3 from) the y-coordinate of each point. This, as well as plotting of the shifted letter, can be done by the
following code:
% Version 1 (presented only to illustrate the basic concept;
% its steps should NOT be mimicked)
% Before running this code,
% execute the code for part 1 and do NOT clear the workspace.
shift_horiz = 0.8;
shift_vert = -0.3; % define the shift’s components
x_letter_shifted = x_letter + shift_horiz;
y_letter_shifted = y_letter + shift_vert;
% shift the coordinates of the letter
figure(4021);
plot(x_letter,y_letter,x_letter_shifted,y_letter_shifted,’k’)
axis([0 2 -0.5 1])
% plot the original (default blue) and the shifted (black) letters
% inside the rectangle [0 2] x [-0.5 1]
A note is in order about Matlab’s syntax used here. Above, x_letter and y_letter are row vectors of length
505. (Why so? 5
代写 MATH 122作业、代做R编程设计作业、代写LINEAR ALGEBRA作业、R实验作业代写
) On the other hand, shift_horiz and shift_vert are scalars. By rules of Linear Algebra,
you cannot add a scalar to a vector. Yet, Matlab allows one to do so, interpreting this operation as the addition
of the same scalar to each entry of the vector. However, this is where Matlab’s loose interpretation of the matrix
addition ends. Namely, it will not allow you to add two vectors of different lengths, or add a matrix to a vector. In
particular (see below), it will not allow you to add a 2 × 505 matrix and a 2 × 1 column vector.
The above code does the job of shifting the letter. However, for Parts 2 – 3 of this Project, it will be convenient
to represent your letter as a single matrix, rather than working separately with its x- and y-coordinates (i.e.,
x_letter and y_letter, respectively). To this end, let us consider the first point of the letter. In the xy-plane,
it is represented by the column vector
(
x letter(1)
y letter(1) )
. Likewise, matrices
(
x letter(1) x letter(2)
y letter(1) y letter(2) )
and (
x letter(1) x letter(2) x letter(3)
y letter(1) y letter(2) y letter(3) )
represent the first two and three points of the letter, etc. Generalizing, matrix
letter matrix = (
x letter
y letter )
represents the entire letter. What are its dimensions? Again, do not proceed without answering this question: it
will come back and bite you. To check your answer, you may type size(letter_matrix) after running the
code shown below.
5Do not read on until you answer this question; otherwise you will likely run into problems later on. If you are not sure, review
the code in Part 1, paying attention to its first command line. If you do not understand what that line does, follow the suggestion in the
paragraph before the Note at the end of the Introduction.
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The following code (Version 2) will produce the same result as Version 1 above, but will handle the letter as a
matrix rather than two separate rows of x- and y-components:
% Version 2 (the version to be mimicked)
% Before running this code,
% execute the code for part 1 and do NOT clear the workspace.
% Step 1:
shift_horiz = 0.8;
shift_vert = -0.3; % define the shift’s components
shift_vector = [shift_horiz; shift_vert];
% define the vector of the shift
shift_matrix = repmat(shift_vector, 1, length(x_letter));
% see the end of the Introduction
% Step 2:
letter_matrix = [x_letter; y_letter];
% create the matrix containing the coordinates of the letter
% Step 3:
letter_matrix_shifted = letter_matrix + shift_matrix;
% shift the letter
% Step 4:
x_letter_shifted = letter_matrix_shifted(1,:);
y_letter_shifted = letter_matrix_shifted(2,:);
% these commands extract the 1st and 2nd rows of the shifted
% letter matrix and put them into two separate row-vectors
figure(4022);
plot(x_letter,y_letter,x_letter_shifted,y_letter_shifted,’k--’)
axis([0 2 -0.5 1])
% plot the original (default blue) and the shifted (black, dashed)
% letters inside the rectangle [0 2] x [-0.5 1]
In the assignments to follow, you should use Step 2 of this code without change and use Steps 3 and 4 after
appropriate modification.
To complete this example, let us recall (see Example 4b in the lecture for Sec. 3.7) that translation of a vector
is not a linear transformation. Above, we have translated not a vector but a matrix. However, this matrix is just a
collection of column vectors representing individual points of the letter. Therefore, translation of this matrix is also
not a linear transformation.6 You will be asked if the transformation that you need to perform in Part 2 is linear or
not.
Let us now return to the assignment of the Project.
Part 2 Write a Matlab code that reflects your letter about the y-axis using a matrix–matrix multiplication. One of
the matrices is your letter’s matrix; the other one must be a matrix derived by you in Project 1. Also, your
work must satisfy the following requirements.
Note 1: You must base your code on Version 2 of the code in the Example above.
6There is a trick, commonly used in computer graphics, of how to make a translation appear as a linear transformation. Google
‘homogeneous coordinates’ if you are curious. We, however, will not consider this trick in this Project.
4
Note 2: You must give your counterpart of the shift_matrix from the Example a different name,
which would be more descriptive of the action of that matrix. Indeed, you are not shifting your letter in this
exercise, right? So, the name of the matrix should reflect what it does to your letter. Your score will be
reduced if the name that you give to the matrix does not reflect its action.
Note 3: While plotting the original and transformed letters in the same figure, you must use different line
styles (see help plot) for the two letters.
Is this transformation linear? Justify your answer by referring to an appropriate material in the textbook or
in the class notes.
Part 3 Here you will explore the other way of defining a linear transformation, when you do not know its matrix
but do know how it transforms a basis. Follow the steps listed below to reflect your letter about a line L:
y =(tan(π/6))x.
(a) A convenient basis is formed by the vector along line L and by a vector perpendicular to it:
v1 = [cos(pi/6); sin(pi/6)];
v2 = [-sin(pi/6); cos(pi/6)];
Note that v1
and v2
form an orthonormal basis.
On a piece of paper, sketch line L and vectors v1
and v2
. Also, sketch, and then record, the result of their
reflection about line L:
Tv1 = % this should be the result of reflection of v1 about L
Tv2 = % this should be the result of reflection of v1 about L
(b) Find coordinates of each column of letter_matrix in the basis {v1
, v2}. Use the fact that this
basis is orthonormal and thus use a formula from Sec. 3.6 for finding coordinates in such a basis.
You can avoid doing the work column by column and instead do it in one step using matrix multiplication:
c1 = some_matrix_A * some_matrix_B;
c2 = similar
Here c1 and c2 are 1×505 row vectors containing the coordinates of each column of letter_matrix in
the basis {v1
, v2}. One of the “some_matrices” is a basis vector and the other is the letter_matrix.
Note that one of them must be transposed. If you answered the questions about dimensions when working
through the Example before Part 2, you should be able to easily identify these “some_matrices”.
(c) Based on the formula
T(v1c1 + v2c2) = T(v1)c1 + T(v2)c2,
write a formula of the form:
letter_matrix_reflected = (Transformed vector 1)*(coordinates 1) + ...
(Transformed vector 2)*(coordinates 2);
% "..." at line’s end means that current command continues on next line.
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where the terms on the right-hand side come from your work in previous steps. Now, mimic Step 4 of the
code in Example before Part 2 to plot the reflected letter. Show the original and reflected letters in the same
plot.
Finally, to help you see that your reflected letter is indeed a reflection of the original letter about line L, add
that line to your plot. You can do so in many ways; for example, plot a vector yL=xL*tan(pi/6) versus
vector xL=[0 1]. Similarly, plot vectors {v1
, v2} in the same figure; see help plotv.
Note 1: In order for your plot to look true to its description above, you need to use one of the two options,
axis(’square’) or axis(’equal’) after your plot. Type help axis in Matlab for an explanation.
Whichever option you use, your vectors {v1
, v2} must look perpendicular in your plot.
Note 2: Do not use different colors for your letters. These are (almost) impossible to tell apart in a blackand-white
printout. Instead, use either different line styles or line widths; see examples under help plot.
Credit for any of the Bonus parts below will be given only if your work for that part is more than 60%
correct and clearly presented.
Moreover, in the past, I had instances where students would submit hard copies of extra-credit plots that looked
correct, but then when I would try to run their codes, the codes would either not run at all or produce different
plots. Therefore, make sure that your extra-credit codes run with a single click of the “Run” button in Matlab’s
Editor. If I find this not to be the case, I will reduce your score by 5 points instead of giving you extra credit.
Bonus
for Part 3 In a separate figure, plot c2 versus c1. Similarly to how a plot of y_letter versus x_letter shows
you the letter in the natural basis, the c2 versus c1 plot shows you the letter in the basis {v1
, v2}. I.e.,
this is how the letter will look like to someone for whom the latter basis is natural. Explain the appearance
of this plot relative to the original plot of your initial.
Hint: Consider how the basis {v1
, v2} is related to the natural basis {e 1
, e 2}. Compare this to how the
two plots are related.
Credit will be given only for a correct explanation, not for a plot or for answering the prompts in the Hint.
Bonus–1 Write a Matlab code that rotates your letter about its right-most point by ? = π/6. Follow these steps.
(i) Translate the letter so that its right-most point is at the origin.
(ii) Rotate the shifted letter about the origin. The transformation matrix that you will need was derived by
you in Project 1; see also p. 236 in the textbook.
(iii) Translate the result of Step (ii) so as to undo the shift made in Step (i).
Note: Plot the original and transformed letters together, using different line styles. Make sure to adjust the
scale of your axes (see Part 3) so that the transformed letter looks exactly as it description suggests.
Is this transformation linear? In justifying your answer, follow the lines of Example 5 in the notes for Section
3.7. You may also review the answers in the Example above.
Bonus–2 In the very first lecture of this course, I told you that all computer graphics (in particular, that employed in
movies and video games) is based on Linear Algebra. That is, the moving objects that you see on the screen
are produced by linear and nonlinear transformations applied to vectors and matrices. In the aforementioned
lecture I used a waving flag as a simple example of a moving object. In this Bonus problem, since you
6
already have a code producing your initial letter, I will ask you to make this letter, rather than a flag, wave.
To preview what kind of results you may get after doing this Bonus part, type ‘waving flag animation’ into
Google.
We will proceed as follows. First, I will present an example of how to produce a single plot of a letter with
wavy contours. This example will consist of two steps. Then, you will be asked to modify my code to make
these contours wave (i.e., move in a periodic, oscillatory manner) in time.
Example
To begin, run the code that produced your letter in Part 1 and do not clear the workspace.
To get the very basic understanding of the result of a nonlinear transformation, run the following code:
x_letter_waveall=1.2*x_letter+0.02*sin(20*x_letter)-0.01*cos(30*y_letter);
y_letter_waveall=0.9*y_letter+0.015*sin(25*x_letter+35*y_letter)+...
0.04*cos(15*x_letter-20*y_letter);
figure(4777);
plot(x_letter_waveall,y_letter_waveall,’k’)
axis([-0.1 1.2 -0.1 1.2])
The lines of the letter are now curved, not straight as before. This is because by adding terms proportional
to the nonlinear functions sine and cosine, we changed coordinates of each line in a highly nonlinear way.7
The above code changes all lines of the letter simultaneously. This is not what you would expect to happen
to a flag. Rather, you would expect that its flagpole maintains its straight shape at all times while the banner
at any time moment has a wavy shape. To produce such a picture, you would have to selectively modify only
those parts of the flag which can move.
To illustrate the idea, below I show an excerpt of the code that makes only the top and right sides of the letter
(i.e., segments 2 and 3) wavy while keeping the rest of it intact.
% Setup of segment 1 is the same as before.
xmin_2=0.7; ymin_2=0.05; xmax_2=0.7; ymax_2=0.95;
x_2=xmin_2+(xmax_2-xmin_2)*t; y_2=ymin_2+(ymax_2-ymin_2)*t;
% 2nd segment before it is made wavy
ax2=0.03; % amplitude of waves of x-components of segment2
fx2=20; % spatial frequency of these waves;
% spatial period = 2*pi/fx2
x_2_wavy=x_2 + ax2*sin(fx2*(y_2-ymax_2));
% x-component of the wavy segment2
% Its form will be commented on after the code.
y_2_wavy=y_2; % For simplicity we assume that the y-coordinates
% of segment2 don’t move.
xmin_3=0.7; ymin_3=0.95; xmax_3=0.45; ymax_3=0.8;
x_3=xmin_3+(xmax_3-xmin_3)*t; y_3=ymin_3+(ymax_3-ymin_3)*t;
% 3rd segment before it is made wavy
7As mentioned in the Introduction, nonlinear transformations make straight lines into curved ones because they change different line
segments differently. Indeed, each segment of your letter is just a set of points on the screen. That is, that segment consists of many
subsegments joining pairs of its adjacent points. A nonlinear transformation transforms each of these subsegments slightly differently; for
example, it rotates the subsegment joining points 1 and 2 slightly differently than the subsegment joining points 2 and 3, etc. This makes
the nonlinearly transformed segments of the letter become curved.
7
ax3=0.04; fx3=50; ay3=0.02; fy3=45;
% ampl’s and spatial freq’s of waves on segment3
x_3_wavy=x_3 + ax3*sin(fx3*(y_3-ymin_3));
y_3_wavy=y_3 + ay3*sin(fy3*(x_3-xmin_3));
% x- and y-components of the wavy segment3
xmin_4=x_3_wavy(end); ymin_4=y_3_wavy(end);
% The upper endpoint of segment 4 is made to move
% according to the motion of segment3.
xmax_4=0.45; ymax_4=0.65;
x_4=xmin_4+(xmax_4-xmin_4)*t; y_4=ymin_4+(ymax_4-ymin_4)*t;
% 4th segment
% Setup of the 5th segment and plotting of the entire letter
% are the same as before.
Note that (for simplicity), we did not change the top point of segment 2. In fact,
x_2_wavy(end)=x_2(end)=xmax_2 because y_2 is defined so that y_2(end)-ymax_2=0.
Similarly, the top point of segment 3 is not changed either.
From this simple example you may appreciate some difficulties occurring in producing animated graphics.
Namely, one has to make sure that end points of connecting segments always move in the same way. This
was easy to do for segments 2 and 3, and also for segments 3 and 4 (see the code), but would have been
much harder if we had tried to make the top end point of segment 1 to lie exactly on the curved segment 2.
Moreover, recall that the above code only produces a snapshot of a wavy flag at just one instance in time.
An animation will need to wrap a loop around such a code in order to make the picture change over time.
This is your main task in this assignment.
Specifically, your task is to modify the above code so as to make at least two, but not all, segments of
your initial letter wave, exhibiting two or three full cycles (i.e., periods) of oscillations. In at least one of the
segments, both x- and y-components are to vary nonlinearly in time.
Please take into account the following suggestions and hints.
What functions of time exhibiting periodic oscillations do you know?
You will need to use Matlab’s for-loop, which will allow you to specify the duration of the waving.
Model the simplest form of waving, whereby only the amplitude, but not spatial frequency, of the
waves will change in time.
When I run your code, I should be able to see a movie of your waving initial letter on my screen, more
or less in real time. If you find that it runs too fast, consider placing a pausing command (something
like pause(0.1)) after the plotting command; see help for pause.
You (and I) should be able to change the duration of the waving by changing just one parameter of the
for-loop.
Acknowledgment:
The idea of drawing one’s initial is borrowed from Sec. 3.1.1 of R.C. Penny, “Linear Algebra: Ideas and Applications”
(Wiley-Interscience, Hoboken, 2004).
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APPLIED LINEAR ALGEBRA, MATH 122 Answer Sheet for Project 4
Submission instructions:
1. There will be a significant number of plots and printouts (of codes) in this project. In order both to organize
your work and to save paper, please put all of your printouts and plots8
into a Word file and print only that
file. If this is not done, I will reduce your score by 5 points. Append any handwritten work to it as needed.
2. In addition to the printouts of your codes, you must also e-mail me these codes. These codes should produce
the action required in the assignment when I run them. For each code which you submit as either a hard or
soft copy but not both, I will reduce your score by 3 points.
The subject line of your e-mail must contain the string MATH_122. It is case-insensitive; thus, math_122
or Math_122 will also work. However, the underscore must be present: e.g., math122 or math 122 will not
work. Moreover, the subject line must contain your name(s) and indicate the fact that this is a submission
of codes for Project 4. I will reduce your score by up to 4 points if this is not done.
3. Your codes must be named according to this convention: p4_YourName(s)_exExercise#.m. E.g.,
my own code for Exercise 2 would be named p4_tlakoba_ex2.m. For each code whose name does not
follow this convention, I will reduce your score by 2 points.
4. Finally, a strong request: Contrary to what you might have learned in other classes, I ask you not to put
commands clc and close all at the beginning of your codes. If I want to clear my command window
and close figure windows, I will do so myself, without unsolicited outside “help”.
Name(s):
Part 1 (27 points)
Include the plot and the code into the Word document and also e-mail the code to me.
Part 2 (32 points)
Include the plot and the code into the Word document and also e-mail the code to me. Make sure you label (e.g.,
by hand) the original and the transformed letters in the plots.
Is this transformation linear? Please explain referencing a corresponding statement in the notes or in the first
half of Sec. 3.7.
8You will need to create a .png, .jpeg, or .tiff figure to include into a Word document. The syntax in Matlab is: print -dpng ’foldername/filename’;
see help print for more details.
9
Part 3 (41 points)
Include the plot and the code into the Word document and also e-mail the code to me.
Also, make sure to attach the hand-drawn sketch of line L and vectors v1, v2, T(v1), T(v2).
Bonus for Part 3 (5 points)
Include the plot and the code into the Word document. You may answer the question either here or in that document.
Bonus–1 (23 points)
Include the plot and the code into the Word document and also e-mail the code to me.
Is this transformation linear? Please explain.
Bonus–2 (31 points)
Include the plot and the code into the Word document and also e-mail the code to me.
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