E7 Fall 2019, UC Berkeley
Homework Assignment 9
Induction & Linear Alge
a
The purpose of this lab is to introduce you to induction and linear alge
a in MATLAB.
Some commands that may be helpful are: rank, spdiags, ones, reshape, pinv.
Note: Use the templates provided (HW9 Template.m) in the assignment download to com-
plete this assignment. The template has a very specific format to enable auto-grading, and if
the format is altered your assignment may not be properly graded (this will be apparent to
you). Please also notice that variables in BOLD FONT will be checked to calculate you
grade. In addition, the template file already reflects much of the content of the assignment.
The due time is 11:59am (noon) November 15, 2019. NO LATE HOMEWORK
WILL BE ACCEPTED. Please upload the following files through the bCourses website:
• HW9 solution.m
• indplot.m, indplot2.m, linearequationtest.m, rotate2d.m, rotate3d.m,
inverserot.m, pet weights.m, avgA
ay.m, gaussJ.m
• SummationPlot.png, FunctionPlot.png, 2DSquareRotated.png,
3DPlaneRotated.png, InverseRotation.png
• HW9 solution.pdf (created using the publish command)
Directions to upload files can be found here.
1. Induction. You can read more about Induction here.
(a) Given the series
S1 =
N∑
n=1
(2n− 1), S2 =
N∑
n=1
n2,
create the function indplot
function [S1,S2] = indplot(N)
The function indplot takes an integer N as its input and calculates the summation
S1 and S2 according to the above equations. It also generates one plot with two
curves showing how each series varies as a function of the integer n, when n ranges
from 1 to N. Save the plot as a .png with the print command, with the filename
’SummationPlot’.
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https:
en.wikipedia.org/wiki/Mathematical_induction
E7 Fall 2019, UC Berkeley
Note:You should use the plot to better understand the relationship between the
expression on the left and right hand side of the equation.
(b) Create a function indplot2.m which plots the following expressions for the same
values of N as in part (a) on a new figure:
f1(N) = N 2, f2(N) = N(N+1)(2N+1)6 .
With the following function header, and f1 is the a
ay of values of f1(N) above,
and f2 is the a
ay of values of f2(N) above.
function [f1,f2] = indplot2(N)
Evaluate indplot and indplot2 at N = 50 in your main script to view the plots.
From a comparison of the plots, you should confirm the following identities fo
any N = 1, 2, ...
N∑
n=1
(2n− 1) = N 2,
N∑
n=1
n2 =
N(N + 1)(2N + 1)
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(c) Note: This following part is good practice, but will not be graded for this home-
work assignment. Now that you have used MATLAB to prove the mathematical
induction, try proving these identities by hand for N = 1, 2, ...
2. Consider the system
Ax = b (1)
Write a function with the following function declaration line
func t i on message = l i n e a r e q u a t i o n t e s t (A, b)
that will take A and b as inputs and output one of the following messages (stored in
your main script as the variable message) depending on the existence and uniqueness
of the solution of the system.
There exists a solution to the system if the rank of the matrix A is equal to the rank
of a the matrix A augmented with the right hand side solution vector b (you can read
more about the rank of matrices here).
The solution to the system is unique if the number of variables in the system is equal
to the rank of the matrix A.
– ’Solution exists and is unique.’
– ’Solution exists but is not unique.’
– ’Solution does not exist.’
A couple example systems are listed below.
2
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www.mathworks.com/help/matla
ef
ank.html
E7 Fall 2019, UC Berkeley
(a) x1 + 3x2 + 2x3 = 1
2x1 + x2 + x3 = 0
−8x1 + x2 − x3 = 1
Should return message = ’Solution does not exist.’
(b) x1 + 3x2 + 2x3 = 1
2x1 + x2 + x3 = 0
x1 + 8x2 + 10x3 = 1
Should return message = ’Solution exists and is unique.’
(c) x1 + 3x2 + 2x3 = 4
2x1 + x2 + x3 = 3
−8x1 + x2 − x3 = −7
Should return message = ’Solution exists but is not unique.’
Note: If a linear system has a solution that exists but is not unique, it cannot be
solved with the backslash operation. Instead you must use pinv which returns the
pseudoinverse of a matrix in the case where the inverse of the matrix A does not exist.
You can read more about the pseudo-inverse here.
3. One useful application of linear alge
a in engineering and the sciences is the rotation of
shapes, images, objects, etc.. You can find a lot of helpful information for this problem
here.
(a) In this problem we will consider a rectangle with the following Cartesian coordi-
nates, (x,y), for its vertices: (1,1) (4,1) (4,4) & (1,4). Store the coordinates in a
2× 4 matrix C, where the first row is the x coordinates and the second row is the
y coordinates. Then write a function with the function declaration line:
function [D,R] = rotate2d(C,theta)
where theta is the rotation angle (in degrees), and the system of linear equations
is:
x′ = x cos θ − y sin θ
y′ = x sin θ + y cos θ
R is the ’rotation matrix’ which is described by the system above. D is the
new (x,y) coordinates, denoted in the equations as x′ and y′. The system can be
described in the matrix form as D=R*C.
Rotate the rectangle 45 degrees. Plot the original shape and the rotated shape,
with the command fill. Label your axes and save the plot as a .png with the
print command, with the filename ’2DSquareRotated’.
(b) Say the original rectangle given in the problem is now a plane in 3 dimensions, with
coordinates in (x,y,z). The coordinates are (1,1,0), (4,1,0), (4,4,1), and (1,4,1).
Store the coordinates in a 3 × 4 matrix E where each row co
esponds to the x,
3
https:
www.mathworks.com/help/matla
ef/pinv.html
http:
mathworld.wolfram.com/RotationMatrix.html
E7 Fall 2019, UC Berkeley
y, and z coordinates respectively (like in part a). Then write a function with the
declaration line:
function [F,Rx,Ry,Rz] = rotate3d(E,alpha,beta,gamma)
where alpha is the x-axis rotation angle, beta is the y-axis rotation angle, gamma
is the z-axis rotation angle, and the system of linear equations for each direction
are given as follows. The angles should be in degrees.
For Rotation about X:
x′ = x
y′ = y cosα− z sinα
z′ = y sinα + z cosα
For Rotation about Y:
x′ = x cos β + z sin β
y′ = y
z′ = −x sin β + z cos β
For Rotation about Z:
x′ = x cos γ − y sin γ
y′ = x sin γ + y cos γ
z′ = z
Rx, Ry, and Rz are the ’rotation matrices’ described by the systems above. F is
the new (x,y,z) coordinates, denoted in the equations as x′, y′, and z′.
Rotate this plane, first 30 degrees about the Z axis, second 60 degrees about the
Y axis, and finally 45 degrees about the X axis. Plot the original shape, and the
otated shape using the command fill3. Label your axes and save the plot as a
.png with the print command, with the filename ’3DPlaneRotated’.
(c) Say instead you are the shape in it’s final position and you want to find it’s original
position. Using the relations above, and a square with (x,y) coordinates: (1,0),
(0,1), (-1,0), and (0,-1). Store the coordinates in a 2 × 4 matrix XYnew, where
the first row is the x coordinates and the second row is the y coordinates. Then
write a function with the function declaration line:
function [XYold] = inverserot(XYnew,theta)
Find what this squares original position, stored in the variable XYold, was if it
had been rotated 45 degrees(the angle should be stored in the variable theta).
Plot the original shape and the rotated shape, with the command fill. Label
your axes and save the plot as a .png with the print command, with the filename
’InverseRotation’.
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E7 Fall 2019, UC Berkeley
4. Suppose the figure below depicts a network of one-way roads, with a
ows indicating
the direction of traffic flow. The number of vehicles entering and leaving the network
(per hour) at nodes A, B, C and D are shown on the figure and x1, x2, x3 and x4
denote the number of vehicles (per hour) passing through the
anches AD, DC, CB
and AB respectively. Suppose the system is in equili
ium, that is, no new vehicles
appear within the network and none are lost. Thus, at each node, the number of
vehicles coming towards the node is equal to the number of vehicles moving away from
the node.
Figure 1: Problem 4
(a) Write down the equations governing the above system. Read the paragraph ex-
plaining the system carefully in order to write the equations.
(b) Build a linear system Ax = b by transforming the equations you wrote down in
part a. Use the variable SYSA for 4×4 matrix A, and the variable veca for vecto
(4 × 1 that represents the amount of cars entering and leaving the network pe
hour). Each row of SYSA should co
espond to the equations at each node A,
B, C and D. Each column of SYSA should co
espond to x1, x2, x3 and x4.
(c) Use your function from problem 2 to determine the existence and uniqueness of
the solution of the system, save the message in the variable solution4a. Then
solve for x if possible, save the result in variable vpha. If the solution exists, but
is not unique remember to use the function pinv in order to find a solution to the
problem. Also, if the solution does not exist return vpha as an empty a
ay [ ].
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E7 Fall 2019, UC Berkeley
(d) Suppose the
anch AB is closed for maintenance, so you know x4 = 0. Create
a new linear system of equations by adding an additional equation to the linea
system you created in part a, use the variable SYSB for the matrix A and vec
for the vector b. Determine the existence and uniqueness of the solution of the
new system, using your function from problem 2