You are not expected to finish the entire quiz in the allotted time. It has been designed to be reasonably challenging so that, among other things, everyone in the class gets a good Matlab workout.
Hint: Don't forget about the linspace command. If you didn't remember it enough to forget it, use
> help linspace
to refamiliarize yourself with it.
Note that these instructions are identical to the ones for Quiz 2, except for the specification of the file that you must create to complete the quiz. In particular, the name of the startup script is again
q2_setup
I am repeating the instructions here so that the quiz is self-contained.
All of the commands that you use to complete this quiz must be entered in the file
/phys210/$LOGNAME/matlab/q3.m
The first statement in this file must be
q2_setup
If this line is not included you will be unable to complete
the quiz. Also, if you prefer to work the questions interactively and
then transfer your answers to q3.m be sure to type
>> q2_setup
before you do anything else.
q2_setup is a script that defines 13 variables: 8 row vectors
v1, v1_el, m1_row, m1_col, w1, w2, w3 and w4
and 5 matrices
m1, a1, a2, a3, a4
You can display any of these by simply entering their names at the
Matlab prompt, but be warned that v1 and m1 contain many
elements. I'll leave it as an exercise for you to determine how
many.
Try to answer the questions using as few Matlab statements as possible, but note that in several cases it may be advantageous to first copy a vector or matrix and then modify it, i.e,
>> foo1 = foo
>> foo1(...) = ...
I will use (copy) to label the instances where I suggest you do this.
I will indicate the number of statements you are expected to use in the parenthesis following the question label.
IMPORTANT: Several questions ask you to create vectors or matrices by selecting subsets of elements other vectors and matrices. Unless specified otherwise, the elements in the new arrays should be in the "same order" as those in the original.
Example 1: Given
a = 1 2 3 4 5 6 7 8 9 10
the vector
a1 = 3 9 10
is a subset of elements of a that are in the "same order" whereas
a2 = 9 3 10
is not.
Example 2: Given
m =
10 15 20 25
30 35 40 45
50 55 60 75
the matrix
m1 =
10 25
50 75
is a subset of elements of a that are in the "same order" whereas
m2 =
20 15
35 40
is not.
I will denote the questions where the order of elements is to be preserved in this manner with (order)
OK, on with the quiz, and remember that the first line in q3.m, or
the first command that you type should you wish to first solve the
problems interactively, must be
q2_setup
a1, a2, a3 and a4 are all 4 x 4 matrices. Use them
to construct an 8 x 8 matrix m7 having the structure:
\[ \left[ \begin{array}{cc} a_1 & a_2 \\\ a_3 & a_4 \end{array}\right] \]
Denoting the 4 x 4 zero and identity matrix by 0 and 1, respectively,
construct a 12 x 8 matrix m8 that has the structure:
\[ m_8 \equiv \left[ \begin{array}{cc} {\bf 0} & a_2 \\\ a_3 & a_4^T \\ a_1 & {\bf 1} \end{array}\right] \]
where \( a_4^T \) denotes the transpose of \( a_4 \).
Construct the following vector using a single assignment statement:
\[ v_7 \equiv \left[ \begin{array}{c} 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right] \]
Assign to the variable v7.
Construct the following vector using at most two assignment statements:
\[ v_8 \equiv \left[ \begin{array}{c} 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 8 \end{array}\right] \]
Assign to the variable v8.
Construct the following matrix using a single assignment statement: \[ m_9 \equiv \left[ \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{array}\right] \]
Assign to the variable m9.
Define a column vector m1_md that contains every 10th element of the
main diagonal of m1, starting with the first element of the diagonal.
Construct a matrix m10 which has a main diagonal given by every
11073-rd element of v1, starting with the 2001-st element, with 0 elements
everywhere else.
Use a single Matlab statement to produce the following matrix:
\[ m_{11} \equiv \left[ \begin{array}{ccccccc} 0.0500 & 0.0600 & 0.0700 & 0.0800 & 0.0900 & 0.1000 & 0.1100 \\ 0.1200 & 0.1300 & 0.1400 & 0.1500 & 0.1600 & 0.1700 & 0.1800 \\ 0.1900 & 0.2000 & 0.2100 & 0.2200 & 0.2300 & 0.2400 & 0.2500 \\ 0.2600 & 0.2700 & 0.2800 & 0.2900 & 0.3000 & 0.3100 & 0.3200 \\ 0.3300 & 0.3400 & 0.3500 & 0.3600 & 0.3700 & 0.3800 & 0.3900 \\ 0.4000 & 0.4100 & 0.4200 & 0.4300 & 0.4400 & 0.4500 & 0.4600 \\ 0.4700 & 0.4800 & 0.4900 & 0.5000 & 0.5100 & 0.5200 & 0.5300 \\ 0.5400 & 0.5500 & 0.5600 & 0.5700 & 0.5800 & 0.5900 & 0.6000 \\ 0.6100 & 0.6200 & 0.6300 & 0.6400 & 0.6500 & 0.6600 & 0.6700 \\ 0.6800 & 0.6900 & 0.7000 & 0.7100 & 0.7200 & 0.7300 & 0.7400 \end{array}\right] \]
Assign to the variable m11.
Use 3 Matlab statements, including a copy, to make a matrix m12 in
which the upper left 3 x 3 submatrix of m11 has been replaced with
the 3 x 3 identity matrix, and the lower right 4 x 4 submatrix has
elements:
\[ \left[ \begin{array}{cccc} 16 & 15 & 14 & 13 \\ 12 & 11 & 10 & 9 \\ 8 & 7 & 6 & 5 \\ 4 & 3 & 2 & 1 \end{array}\right] \]
Evaluate the following function at 101 uniformly spaced values of \(x\) ranging from \(-\pi/2\) to \(\pi/3\) inclusive.
\[ \sin\left(x^3\right) + \frac{\cos\left(\sqrt{|x|^4}\right)}{x+1} + 6 \]
Assign the result, which should be a row vector of length
101, to the variable fv.
You should first create a row vector with a name of your
own choosing (other than a name that is used in one of the other
questions in the quiz) to store the values of \(x\).
Set up and solve the following linear system:
\[ \begin{eqnarray} 78 v + 17 w + 4 y + 88 z &=& 64 \\ 44 v + 85 w + 11 z &=& 42 \\ 3 v + 15 w + 48 y + 37 z &=& 62 \\ 12 v + 28 w + 80 y + 98 z &=& 57 \\ \end{eqnarray} \]
Assign the solution vector (a column vector)
\[
\left[\begin{array}{c}
v \\ w \\ y \\ z
\end{array}\right]
\]
to x_44.
Choose variable names of your own for the coefficient matrix and right hand vector of the system.
Find the maximum and minimum values of the vector v1. Assign
to the variables v1_max and v1_min respectively.
Use the sort command (and some other piece of syntax) to find the
maximum value of the vector cos(v1)
without using the max function. Assign to the variable
cos_v1_max.
Find the average (mean) and median values of the matrix m1. Assign
to the variables m1_mean and m1_median respectively, and note that
both answers should be a single scalar value (i.e. not
a vector).
Compute \(17!\) using an assignment statement with exactly 10
non white-space characters on the right hand of the = sign. Assign
to fact_17 (the value will be not be exact, as it is in Maple).
How many elements in m1 are equal to 23? Assign the value to n23.
This can be done with one statement using nested function calls, but you certainly aren't expected to know how to do it at this point.
Try to resist using google, though.
Hint: The Matlab relational operator for equality is == and
Matlab uses the values 1 and 0 for Boolean (logical) true and false,
respectively. Try evaluating
2 == 2
and
3 == 2
IMPORTANT
When you've completed the quiz, execute the following from the terminal command line:
% date > /phys210/$LOGNAME/matlab/date
AS USUAL, YOU CAN THEN WORK ON THE HOMEWORK OR TALK TO ME ABOUT YOUR TERM PROJECTS WHILE YOU WAIT FOR THE CLASS TO FINISH THE QUIZ