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MATLAB Homework
- Matlab Assignments for Calculus at Ohio
- A Very Brief Intro to Matlab ( PDF ) ( LaTex )
- Entering Formulas in Matlab ( PDF ) ( LaTex )
- Matlab Commands for Linear Algebra ( PDF ) ( LaTex )
- Math 2301 - Calculus I
- Math 2302 - Calculus II
- Math 3300 - Calculus III
- Math 3400 - Differential Equations
- Math 3200 and 3210 - Linear Algebra
- Entering formulas in MATLAB [PDF]
For Instructors: How to Give MATLAB Homework
- MATLAB is now available in computer labs campus-wide and in the residence halls.
- Simple homework assignments have been written for use in 2301, 2302, 3300, 3400 and 3200.
- The undergraduate committee has passed a resolution strongly encouraging use of the assignments in all Calculus sections. Use is to follow the guidelines in this document.
- The chief goal of the assignments is to teach students to think when they compute.
- All the departments in the College of Engineering and Technology have decided to make MATLAB the primary software used in coursework. They consider exposing students to the program in calculus to be a very positive service.
- MATLAB homework is available at the links above.
Using the Homework
- Make the homework no more than 5% of the grade, or extra credit for the same amount.
- Hand out "MATLAB Assignments at Ohio University" and instruct them to download: "A Brief Intro to MATLAB." Instruct them to keep these as references.
- Assign about five assignments per quarter. Assignments can be given as hand-outs, or, you may assign them from the web page.
- Grade the assignments.
- Hand back the assignments and discuss the mathematical content in class .
- Do not attempt to teach the students to use MATLAB. They pick up the basics of MATLAB directly from the assignments. As a corollary, you do not need to know MATLAB well. At most, you need to be able to follow the explicit instructions of the assignments. Almost all student problems will be a matter of not typing the commands exactly as directed.
- The theme of most assignments is to promote the mathematical ideas which are necessary for computing intelligently. Often this involves planting a seed of skepticism in the student's thinking.
- Hints are placed at the bottom of each assignment. The hints are meant to serve as a guide to the main mathematical ideas of each assignment.
- Since use of MATLAB is new at Ohio University, it is best to grade very generously. Usually give full credit as long as the student has touched on all the points in the hint.
- LeTeX versions of the assignments are available on the web page so that you can modify them as you wish.
- To learn more you are invited to read "MATLAB in Calculus and Beyond - An 1804 Fund Project" or the book Technology in Undergraduate Math - A simple Approach or "The MATLAB Workbook -- A Supplement for Calculus, Differential Equations and Linear Algebra". These documents are available here .
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12.5: Systems of Equations Examples and Exercises
- Last updated
- Save as PDF
- Page ID 85166
- Carey Smith
- Oxnard College
By Carey A. Smith
Exercise \(\PageIndex{1:}\) Solving Linear Equations Video and Homework
Watch this 4 minute video. Then code up the example shown and solve using Matlab's inv() function.
(1) Define the matrix A shown in the video.
(2) Define the vertical vector B shown in the video.
(3) X is the unknown vertical vector whose elements are x and y
X = inv(A)*B
Test your answer by computing:
You should get B2 == B
This video shows how to solve a system of equations graphically.
Example \(\PageIndex{1:}\) 2 equations, 2 unknowns Graphic and Matrix methods
1*x + y = 5 -2*x + y = -1
1a. Graphical Solution
Each of 2 equations in 2 unknowns, x and y, can be plotted as a line. Their intersection is a graphical solution of the 2 equations. The equations and their plot is shown here. The intersection is the point (2, 3). So the solution of this system of equations is x=2, y = 3.
1b. Solution using the matrix inverse.
Matrix formulation:
% A*xy = B A = [ 1 1 -2 1] % xy = [x % y] B = [ 5 -1] rank(A) % 2 (full) det(A) % 3 (non-zero) % Method 1: Solution using the inverse of A: % inv(A)*A*xy = inv(A)*B % inv(A)*A = I, so xy = inv(A)*B
% Solution = (2, 3), same as above
1b. Solution using left matrix-division .
Same formulation as in part 1b. The left matrix-division conceptual derivation is:
A*xy = B (1/A)(A*xy) = (1/A)*B [Not B/A--not commutative]
Write (1/A)*B as xy = A\B
% Solution = (2, 3), same as above Solution
The solutions were given in the example.
Important note about left matrix-division:
xy = A\B is “Left Matrix-Division”. It solv es the system of equations by Gaussian Elimination, which is a series of matrix operations. When A is a 3x3 matrix and B is a 3x1 vector, then the result, xy, is a 3x1 vector.
Unfortunately, MATLAB has another function, ldivide(), which sounds like it is the same thing, but it is not. Do not use ldivide(); it does an element-by-element divide, rather than a matrix operation. When A is a 3x3 matrix and B is a 3x1 vector, then the result of using ldivide(), xy is a 3x3 matrix, which is not the solution of the system of equations.
So use either
xy = inv(A)*B
Exercise \(\PageIndex{2:}\) System_of_Equations_HW2
Given this system of equations:
1*x + 1*y + 2*z = 11.1 0*x -1*y + 3*z = 5.4 4*x + 2*y + 1*z = 9.7
Solve for x, y, and z using the following steps:
Part 1 (3 pts). Create the matrix, M, of the coefficients.
Part 2. (1 pt). Create the right-hand side vector of constants. Call it R.
Part 3. (1 pt). Compute the determinant and verify that it is not 0
Part 4. (3 pts) Let xyz = the vector of unknown values of x, y, and z. Solve for xyz using the inverse of M.
Part 5: (3 pts) Solve for xyz using the inverse of M, using left matrix-division.
Part 6: (1 pt) Multiply M*xyz and verify that it equals R.
The answer is not given here.
. This video shows how systems of equations are used to solve a structural engineering truss problem:
Example \(\PageIndex{2:}\) Electric_Circuit_KVL1
You do not need to to be a circuits expert. The equations needed will be given to you. Then these will be put in the form of a system of linear equations that can be solved.
Kirkoff’s Voltage Law:
The sum of the voltages around any loop without a voltage source = 0.
The sum of the voltages around any loop with a voltage source = the voltage of the source.
When a resistor is part of 2 loops, the sum of the currents across the resistor needs to be used. We assume that the current is positive in the direction of the arrows in the drawing. When the direction of the loop arrow is backwards through a resistor, then use the negative of voltage across the resistor.
The voltage supply is V1 = 3 volts.
The resistors are:
R2 = 2 Ohms
R3 = 3 Ohms
R4 = 4 Ohms
The expressions for the voltages across each resistor are:
We use the formula: V = R*I, where V = voltage, R = resistance, I = current.
VR1 = R1*I2 %Voltage across R1 (left to right) VR2 = R2*(I1 - I2) %Voltage across R2 (left to right) VR3 = R3*(I3 - I2) %Voltage across R3 (left to right) VR4 = R4*(I1 - I3) %Voltage across R4 (top to bottom)
The voltages around each loop are:
R2*(I1 - i2) + R4*(I1 – i3) = V1 %Loop1
R1*I2 - R3*(I3 - I2) - R2*(I1 - I2) = 0 %Loop2
R3*(I3 - I2) - R4*(I1 - I3) = 0 %Loop3
Since the unknowns are the currents, I1, I2, I3, we combine like terms, so that we can put this in matrix form:
(R2+R4)*I1 (-R2)*i2 + (-R4)*i3 = V1 %Loop1 (-R2)*I1 + (R1+R2+R3)*i2 + (-R3)*i3 = 0 %Loop2 (-R4)*I1 + (-R3)*i2 + (R3+R4)*i3 = 0 %Loop3 The matrix form is coef*I123 = vconst, where:
coef = [(R2+R4), (-R2), (-R4); (-R2), (R1+R2+R3), (-R3); (-R4), (-R3), (R3+R4)]
I123 = the vector of the unknown currents.
vconst = [V1; 0; 0] The computed value of matrix coef is:
% 6 -2 -4 % -2 6 -3 % -4 -3 7
det(coef) % = 26, so coef is non-singular.
We can now solve for I123 using either the inverse of coef or using left-matrix division.
Once we have computed I123, we can compute the currents in each loop and the voltages across each resistor.
Using the inverse of coef:
I123 = inv(coef)*vconst % 3.8077 % 3 % 3.4615
Using left matrix division: I123 = coef\vconst %Same answer to 5 significant figures
Check the answer by computing the matrix form: check1 = coef*I123
This equals the right-hand-side, vconst.
We can use I123 to compute the currents in each loop:
I1 = I123(1) I2 = I123(2) I3 = I123(3)
We can use these currents to compute the voltages across each resistor:
VR1 = R1*i2 %Voltage across R1 (left to right) VR2 = R2*(I1 - i2) %Voltage across R2 (left to right) VR3 = R3*(i3 - i2) %Voltage across R3 (left to right) VR4 = R4*(I1 - i3) %Voltage across R4 (top to bottom)
Then we can verify that the sums of the voltages around each loop are essentially zero: VLoop1 = -V1 + VR2 + VR4 VLoop2 = VR1 - VR3 - VR2 VLoop3 = VR3 - VR4
Exercise \(\PageIndex{3:}\) Electric_Circuit_KVL2
This circuit is similar to the previous example. R3 has been removed and R5 has been added. The values of the voltage and the resistances have been changed.
The voltage supply is V1 = 6 volts.
R1 = 12 Ohms
R2 = 6 Ohms
R4 = 3 Ohms
R5 = 6 Ohms
Use this formula: V = R*I, where V = voltage, R = resistance, I = current.
VR1 = R1*I2 %Voltage across R1 (left to right) VR2 = R2*(I1 - I2) %Voltage across R2 (left to right) VR4 = R4*(I1 - I3) %Voltage across R4 (top to bottom) VR5 = R5*I3 %Voltage across R5 (top to bottom) The voltages around each loop are:
R2*(I1 - I2) + R4*(I1 - I3) = V1 %Loop1 R1*I2 - R2*(I1 - I2) = 0 %Loop2 R5*I3 - R4*(I1 - I3) = 0 %Loop3
(R2+R4)*I1 (-R2)*i2 + (-R4)*i3 = V1 %Loop1 (-R2)*I1 + (R1+R2+R3)*i2 + (-R3)*i3 = 0 %Loop2 (-R4)*I1 + (-R3)*i2 + (R3+R4)*i3 = 0 %Loop3 The matrix form is coef2*I123 = vconst, where:
coef2 = [(R2+R4), (-R2), (-R4); (-R2), (R1+R2+R3), (-R3); (-R4), (-R3), (R3+R4)]
vconst2 = [V1; 0; 0] Compute the determinant of the coef2 matrix.
Compute I123 using either the inverse of coef2 or using left-matrix division.
Check the computed value of I123 by computing the matrix form: check2 = coef2*I123
This should equal the right-hand-side, vconst2.
Use I123 to compute the voltages across each resistor:
Verify that the sums of the voltages around each loop are essentially zero: VLoop1 = -V1 + VR2 + VR4 VLoop2 = VR1 - VR3 - VR2 VLoop3 = VR3 - VR4
Exercise \(\PageIndex{4}\) Metal Alloy System of Equations
An engineer has samples of 3 Magnesium-Aluminum-Zinc alloys.
These alloys are named Az63, Az92, and Az92 (The 2 numbers represent the percentages of Aluminum and of Zinc)
% The fractions of each metal in each alloy are given in this table: % | AZ63 | AZ91 | AZ92 | % ---|------|------|------| % Mg | 0.91 | 0.90 | 0.89 | % Al | 0.06 | 0.09 | 0.09 | % Zn | 0.03 | 0.01 | 0.02 |
Part 1 (3 pts). Create M3 = 3x3 coefficient matrix with these proportions.
Part 2: (1 pt). Compute the determinant of M3. Since the numbers in the table are small, we expect the determinant to be on the order of 10e-4. For this problem, as long as it is > 10e-8, then the matrix is not singular and can be properly inverted.
Part 3 (2 pts). The engineer wants to combine one sample of each of the 3 alloys, such that the combination has the following weights of each metal (which is written as a vector W3):
W3 = [16.11 % Total weight of Mg in the 3 samples 1.560 % Total weight of Al in the 3 samples 0.330]; % Total weight of Zn in the 3 samples
Create the vector W3 in your code.
Part 4 (3 pts) Let S = vector with the unknown weights of the 3 samples. Solve for S using the matrix inverse method. Part 5 (3 pts) Use inverse or left matrix-division to solve for S using the left-division method.
Part 6. (2 pts) Multply M3 times S and verify that the result = W3
( https://www.azom.com/article.aspx?ArticleID=8675 (Links to an external site.) )
Truss problem
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MATLAB is now available in computer labs campus-wide and in the residence halls. Simple homework assignments have been written for use in 2301, 2302, 3300, 3400 and 3200. The undergraduate committee has passed a resolution strongly encouraging use of the assignments in all Calculus sections. Use is to follow the guidelines in this document.
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always refer to the coordinates in the horizontal and vertical axes respectively. For example, the intensity at the xy coordinates (2, 3) of a MATLAB matrix im is im(3, 2) rather than im(2, 3) . As another example, the MATLAB documentation of meshgrid refers to output variables
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Homework 1 (PDF) Unit 2 Root-Finding. Homework 2 (PDF) Unit 3 Basic Plotting Unit 4 Vectorization. Homework 3 (PDF) Unit 5 Fractals and Chaos. Homework 4 (PDF) Homework 5 (PDF) Homework 6 (PDF) Homework 7 (PDF) Homework 8 (PDF) Unit 6 Debugging Unit 7 Conway Game of Life. Homework 9 (PDF) Homework 10 (PDF)
Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-ineditingand debugging tools, andsupports object-oriented programming. Thesefactors make MATLAB an excellent tool for teaching and research. MATLAB has many advantages compared to conventional computer languages (e.g.,
For instance, to view examples demonstrating plotting in MATLAB, navigate to the MATLAB > Graphics > 2-D and 3-D Plots category and click Examples at the top of the page. To copy the example and supporting files onto your system and open the example, use the buttons at the top of each example: When viewing an example in the Help browser ...
Homework More Projects Videos Course Info Instructor Yossi Farjoun; Departments ... Introduction To MATLAB Programming. Menu. More Info Syllabus The Basics ... assignment_turned_in Programming Assignments with Examples. Download Course.
Introduction to MATLAB. Menu. More Info Syllabus Lecture Notes Assignments Assignments. 6.057 Introduction to MATLAB, Homework 1. Resource Type: Assignments. pdf. 1 MB 6.057 Introduction to MATLAB, Homework 1 Download File DOWNLOAD. Course Info Instructor Orhan Celiker; Departments Electrical Engineering and Computer Science ...
Download the MATLAB Analyze Text Data with String Arrays example to the folder C:\Work\myfiles, and open the sonnets.txt supporting file for that example. Use the supportingFile name-value argument instead of the sfile input argument when the supporting file to open is included in multiple examples or when it has an extension that is not supported by the sfile input argument.
Accepted Answer: Star Strider. I remember that in one of the old Matlab version (2010 or even earlier), in its Help was shown example of the application FFT - IFFT filter to remove noise frequency components of signal, which was close to the sinusoidal. Example was short and useful, but now I need something similar, and can't find it.