In order to successfully complete this assignment you need to participate both individually and in groups during class. If you attend class in-person then have one of the instructors check your notebook and sign you out before leaving class. If you are attending asyncronously, turn in your assignment using D2L no later than 11:59pm on the day of class. See links at the end of this document for access to the class timeline for your section.
![Image showing how matrix multiply works. There is a 2 by 3 matrix multiplied by a 3 by 2 matrix to get a 2 by 2 matrix. The first row in the first matrix is highlighted and the first column of the second matrix is highlighted. The words 'Dot Product' are pointing to the highlighted row and column and the single value output is shown in as the only element in the upper left of the 2 by 2 result. Basically the image is showing that the row [1,2,3] dotted with the column [7,9,11] results in the single output of 58.](https://www.mathsisfun.com/algebra/images/matrix-multiply-a.svg)
Agenda for today's class (80 minutes)¶
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3. Systems of Linear Equations with Many Solutions¶
When we solve a system of equations of the form $Ax=b$, we mentioned that we may have three outcomes:
- a unique solution
- no solution
- infinity many solutions
Assume that we have $m$ equations and $n$ unkowns.
Case 1 $m < n$, we do not have enough equations, there will be only TWO outcomes: no solution, or infinity many solutions.
Case 2 $m = n$, we may have all THREE outcomes. If the determinate is nonzero, we have a unique solution, otherwise, we have to decide the outcome based on the augmented matrix.
Case 3 $m>n$, we have more equations than the number of unknowns. That means there will be redundant equations (we can remove them) or conflict equations (no solution). We may have all THREE outcomes.
We talked about several methods for solving the system of equations. The most general one is the Gauss-Jordan or Gaussian elimination, which works for all three cases. Note that Jacobian and Gauss-Seidel can not work on Case 1 and Case 3.
We will focus on the Gaussian elimination. After the Gaussian elimiation, we look at the last several rows (could be zero) with all zeros except the last column.
If one element from the corresponding right hand side is not zero, we have that $0$ equals some nonzero number, which is impossible. Therefore, there is no solution. E.g.,
$$\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \\ 4 \\ 5 \end{matrix} \right] $$In this case, we say that the system is inconsistent. Later in the semester we will look into methods that try to find a "good enough" solution for an inconsistant system (regression).
Otherwise, we remove all the rows with all zeros (which is the same as removing redundant equations). If the number of remaining equations is the same as the number of unknowns, the rref is an identity matrix, and we have unique solution. E.g., $$\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \\ 4 \\ 0 \end{matrix} \right] \Rightarrow \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \\ 4 \end{matrix} \right] $$
If the number of remaining equations is less than the number of unknonws, we have infinite many solutions. Consider the following three examples:
$$\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \\ 0 \end{matrix} \right] \Rightarrow \left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \end{matrix} \right] \Rightarrow x= [2, 3, x_3]^\top $$where $x_3$ is a free variable.
$$\left[ \begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \\ 0 \end{matrix} \right] \Rightarrow \left[ \begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 3 \end{matrix} \right] \Rightarrow x= [2-2x_2, x_2, 3]$$where $x_2$ is a free variable.
$$\left[ \begin{matrix} 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 5 \\ 0 \end{matrix} \right] \Rightarrow \left[ \begin{matrix} 1 & 2 & 0 & 1 \\ 0 & 0 & 1 & 3 \\ \end{matrix} \, \middle\vert \, \begin{matrix} 2 \\ 5 \end{matrix} \right] \Rightarrow x= [2-2x_2-x_4, x_2, 5-3x_4, x_4]$$where $x_2$ and $x_4$ are free variables.
✅ **QUESTION:** Assume that the system is consistent, explain why the number of equations can not be larger than the number of unknowns after the redundant equations are removed?
Put the solution to the above question here.
✅ **DO THIS:** If there are two solutions for $Ax=b$, that is $Ax=b$ and $Ax'=b$ while $x\neq x'$. Check that $A(cx+(1-c)x') = b$ for any real number $c$. Therefore, if we have two different solutions, we have infinite many solutions.
Put the solution to the above question here.
If $Ax=b$ and $Ax'=b$, then we have $A(x-x')=0$. If $x$ is a particular solution to $Ax=b$, then all the solutions to $Ax=b$ are $\{x+v: v \mbox{ is a solution to the homogeneous system } Av=0\}$.
The solution for $Ax=0$ is always a subspace.
After removing the redundant rows, if the number of equations is the same as the number of unknowns, we have a unique solution. If the difference between this two number is 1, all the solutions lie on a line. If the difference is 2, all the solutions lie on a 2-D plane.
✅ **QUESTION:** What is the solution to the following set of linear equations in augmented matrix form?
$$A = \left[ \begin{matrix} -2 & 4 & 8 \\ 1 & -2 & 4 \\ 4 & -8 & 16 \end{matrix} \, \middle\vert \, \begin{matrix} 0 \\ 0 \\ 0 \end{matrix} \right] $$Put the solution to the above question here.
3. Matrix Multiply¶
✅ **DO THIS:** Write your own matrix multiplication function using the template below and compare it to the built-in matrix multiplication that can be found in numpy. Your function should take two "lists of lists" as inputs and return the result as a third list of lists.
#some libraries (maybe not all) you will need in this notebook
%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing(use_unicode=True)
import random
import time
def multiply(m1,m2):
#first matrix is nxd in size
#second matrix is dxm in size
n = len(m1)
d = len(m2)
m = len(m2[0])
#check to make sure sizes match
if len(m1[0]) != d:
print("ERROR - inner dimentions not equal")
#### put your matrix multiply code here #####
return result
Test your code with the following examples
#Basic test 1
n = 3
d = 2
m = 4
#generate two random lists of lists.
matrix1 = [[random.random() for i in range(d)] for j in range(n)]
matrix2 = [[random.random() for i in range(m)] for j in range(d)]
sym.init_printing(use_unicode=True) # Trick to make matrixes look nice in jupyter
sym.Matrix(matrix1) # Show matrix using sympy
sym.Matrix(matrix2) # Show matrix using sympy
#Compute matrix multiply using your function
x = multiply(matrix1, matrix2)
#Compare to numpy result
np_x = np.matrix(matrix1)*np.matrix(matrix2)
#use allclose function to see if they are numrically "close enough"
print(np.allclose(x, np_x))
#Result should be True
#Test identity matrix
n = 4
# Make a Ransom Matrix
matrix1 = [[random.random() for i in range(n)] for j in range(n)]
sym.Matrix(matrix1) # Show matrix using sympy
#generate a 3x3 identity matrix
matrix2 = [[0 for i in range(n)] for j in range(n)]
for i in range(n):
matrix2[i][i] = 1
sym.Matrix(matrix2) # Show matrix using sympy
result = multiply(matrix1, matrix2)
#Verify results are the same as the original
np.allclose(matrix1, result)
Timing Study¶
In this part, you will compare your matrix multiplication with the numpy matrix multiplication.
You will multiply two randomly generated $n\times n$ matrices using both the multiply() function defined above and the numpy matrix multiplication.
Here is the basic structure of your timing study:
- Initialize two empty lists called
my_timeandnumpy_time - Loop over values of n (100, 200, 300, 400, 500)
- For each value of $n$ use the time.clock() function to calculate the time it takes to use your algorithm and append that time (in seconds) to the
my_timelist. - For each value of $n$ use the time.clock() function to calculate the time it takes to use the
numpymatrix multiplication and append that time (in seconds) to thenumpy_timelist. - Use the provided code to generate a scatter plot of your results.
n_list = [100, 200, 300, 400, 500]
my_time = []
numpy_time = []
# RUN AT YOUR OWN RISK.
# THIS MAY TAKE A WHILE!!!!
for n in n_list:
print(f"Measureing time it takes to multiply matrixes of size {n}")
#Generate random nxn array of two lists
matrix1 = [[random.random() for i in range(n)] for j in range(n)]
matrix2 = [[random.random() for i in range(n)] for j in range(n)]
start = time.time()
x = multiply(matrix1, matrix2)
stop = time.time()
my_time.append(stop - start)
#Convert the lists to a numpy matrix
npm1 = np.matrix(matrix1)
npm2 = np.matrix(matrix2)
#Calculate the time it takes to run the numpy matrix.
start = time.time()
answer = npm1*npm2
stop = time.time()
numpy_time.append(stop - start)
plt.scatter(n_list,my_time, color='red', label = 'my time')
plt.scatter(n_list,numpy_time, color='green', label='numpy time')
plt.xlabel('Size of $n x n$ matrix');
plt.ylabel('time (seconds)')
plt.legend();
Based on the above results, you can see that the numpy algorithm not only is faster but also "scales" at a slower rate than your algorithm.
✅ **QUESTION:** Why do you think the numpy matrix multiplication is so much faster?
Put your answer to the above question here
Congratulations, we're done!¶
If you attend class in-person then have one of the instructors check your notebook and sign you out before leaving class. If you are attending remote, turn in your assignment using D2L.
Course Resources:¶
Written by Drs. Ming Yan and Dirk Colbry, Michigan State University
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.