26.2: Understanding Projections With Code

In this in-class assignment, we are going to avoid some of the more advanced libraries ((i.e. no numpy or scipy or sympy) to try to get a better understanding about what is going on in the math. The following code implements some common linear algebra functions:

#Standard Python Libraries only
import math
import copy

from urllib.request import urlretrieve
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

def dot(u,v):
    '''Calculate the dot product between vectors u and v'''
    temp = 0;
    for i in range(len(u)):
        temp += u[i]*v[i]
    return temp

def multiply(m1,m2):
    '''Calculate the matrix multiplication between m1 and m2 represented as list-of-list.'''
    n = len(m1)
    d = len(m2)
    m = len(m2[0])
    
    if len(m1[0]) != d:
        print("ERROR - inner dimentions not equal")
    result = [[0 for i in range(n)] for j in range(m)]
    for i in range(0,n):
        for j in range(0,m):
            for k in range(0,d):
                result[i][j] = result[i][j] + m1[i][k] * m2[k][j]
    return result

def add_vectors(v1,v2):
    v3 = []
    for i in range(len(v1)):
        v3.append(v1[i]+v2[i])
    return v3

def sub_vectors(v1,v2):
    v3 = []
    for i in range(len(v1)):
        v3.append(v1[i]-v2[i])
    return v3

def norm(u):
    '''Calculate the norm of vector u'''
    nm = 0
    for i in range(len(u)):
        nm += u[i]*u[i]
    return math.sqrt(nm)

def transpose(A):
    '''Calculate the transpose of matrix A represented as list of lists'''
    n = len(A)
    m = len(A[0])
    AT = list()
    for j in range(0,m):    
        temp = list()
        for i in range(0,n):
            temp.append(A[i][j])
        AT.append(temp)
    return AT

Projection function

Do This

Write a function that projects vector \(v\) onto vector \(u\). Do not use the numpy library. Instead use the functions provided above:

\[\mbox{proj}_u v = \frac{v \cdot u}{u \cdot u} u \nonumber \]

Make sure this function will work for any size of \(v\) and \(u\).

def proj(v,u):
    ## Put your code here
    return pv

Let’s test your function. Below are two example vectors. Make sure you get the correct answers. You may want to test this code with more than one set of vectors.

u = [1,2,0,3]
v = [4,0,5,8]
print(proj(u,v))

from answercheck import checkanswer

checkanswer.vector(proj(u,v),'53216508af49c616fa0f4e9676ce3b9d');

Visualizing projections

Do This

See if you can design and implement a small function that takes two vectors (\(a\) and \(b\)) as inputs and generates a figure similar to the one above.

I.e. a black line from the origin to “\(b\)”, a black line from origin to “\(a\)”; a green line showing the “\(a\)” component in the “\(b\)” direction and a red line showing the “\(a\)” component orthogonal to the green line. Also see section titled “Projection of One Vector onto Another Vector” in Section 4.6 on page 258 of the book.

When complete, show your solution to the instructor.

%matplotlib inline
import matplotlib.pylab as plt

b = [3,2]
a = [2,3]

def show_projection(a,b):
    plt.plot([0,a[0]], [0,a[1]], color='black')
    plt.annotate('b', b, 
            xytext=(0.9, 0.7), textcoords='axes fraction',
            arrowprops=dict(facecolor='black', shrink=0.05),
            horizontalalignment='right', verticalalignment='top')
    plt.annotate('a', a, 
            xytext=(0.7, 0.95), textcoords='axes fraction',
            arrowprops=dict(facecolor='black', shrink=0.05),
            horizontalalignment='right', verticalalignment='top')
    plt.plot([0,b[0]], [0,b[1]], color='black')
    
#Finish your code here

    plt.axis('equal')
    
x = show_projection(a,b) ;