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# Part 1 - Scalars and Vectors

For the questions below it is not sufficient to simply provide answer to the questions, but you must solve the problems and show your work using python (the NumPy library will help a lot!) Translate the vectors and matrices into their appropriate python  representations and use numpy or functions that you write yourself to demonstrate the result or property. 

## 1.1 Create a two-dimensional vector and plot it on a graph


```
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

two_d_vector = [.4, 1]

plt.arrow(0,0, two_d_vector[0], two_d_vector[1],head_width=.1, head_length=.1, color ='orange')
plt.xlim(-.2,1)          
plt.ylim(-.2,1.2)
```

## 1.2 Create a three-dimensional vector and plot it on a graph


```
vectors = np.array([[0, 0, 0, 2, 3, 4], 
                    [0, 0, 0, 7, 4, 6],
                    [0, 0, 0, 2, 8, 9]])

X, Y, Z, U, V, W = zip(*vectors)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.quiver(X, Y, Z, U, V, W, length=.5)
ax.set_xlim([0, 4])
ax.set_ylim([0, 6])
ax.set_zlim([0, 5])
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
plt.show()
```


```

```

## 1.3 Scale the vectors you created in 1.1 by $5$, $\pi$, and $-e$ and plot all four vectors (original + 3 scaled vectors) on a graph. What do you notice about these vectors? 


```
from math import e, pi

two_d_vector = [.4, 1]
scaled_to_e = np.multiply(e, two_d_vector)
scaled_to_pi = np.multiply(pi, two_d_vector)
scaled_by_5 = np.multiply(5, two_d_vector)


plt.arrow(0,0, scaled_by_5[0], scaled_by_5[1],head_width=.1, head_length=.1, color ='green')
plt.arrow(0,0, scaled_to_pi[0], scaled_to_pi[1],head_width=.1, head_length=.1, color ='blue')
plt.arrow(0,0, scaled_to_e[0], scaled_to_e[1],head_width=.1, head_length=.1, color ='red')
plt.arrow(0,0, two_d_vector[0], two_d_vector[1],head_width=.1, head_length=.1, color ='orange')




plt.xlim(-.2,3)          
plt.ylim(-.2,6)
```


```

```

## 1.4 Graph vectors $\vec{a}$ and $\vec{b}$ and plot them on a graph

\begin{align}
\vec{a} = \begin{bmatrix} 5 \\ 7 \end{bmatrix}
\qquad
\vec{b} = \begin{bmatrix} 3 \\4 \end{bmatrix}
\end{align}


```
a = np.array([5,7])
b = np.array([3,4])

plt.arrow(0,0,a[0],a[1], head_width=.2, head_length=.2, color='b')
plt.arrow(0,0,b[0],b[1], head_width=.2, head_length=.2, color='r')

plt.xlim(0, 6)
plt.ylim(0,8)

```

## 1.5 find $\vec{a} - \vec{b}$ and plot the result on the same graph as $\vec{a}$ and $\vec{b}$. Is there a relationship between vectors $\vec{a} \thinspace, \vec{b} \thinspace \text{and} \thinspace \vec{a-b}$


```
a = np.array([5,7])
b = np.array([3,4])
c = a - b

plt.arrow(0,0,a[0],a[1], head_width=.2, head_length=.2, color='b')
plt.arrow(0,0,b[0],b[1], head_width=.2, head_length=.2, color='r')
plt.arrow(0,0,c[0],c[1], head_width=.2, head_length=.2, color='y')


plt.xlim(0, 6)
plt.ylim(0,8)
```

## 1.6 Find $c \cdot d$

\begin{align}
\vec{c} = \begin{bmatrix}7 & 22 & 4 & 16\end{bmatrix}
\qquad
\vec{d} = \begin{bmatrix}12 & 6 & 2 & 9\end{bmatrix}
\end{align}



```
c = np.array([7, 22, 4, 16])
d = np.array([12, 6, 2, 9])

c_dot_d = (c*d).sum()

c_dot_d
```




    368




```
np.vdot(c, d )
```




    368



##  1.7 Find $e \times f$

\begin{align}
\vec{e} = \begin{bmatrix} 5 \\ 7 \\ 2 \end{bmatrix}
\qquad
\vec{f} = \begin{bmatrix} 3 \\4 \\ 6 \end{bmatrix}
\end{align}


```
e = np.array([5, 7, 2])
f = np.array([3, 4, 6])

e_cross_f = np.cross(e,f)

e_cross_f
```




    array([ 34, -24,  -1])




```
np.array
```

## 1.8 Find $||g||$ and then find $||h||$. Which is longer?

\begin{align}
\vec{g} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 8 \end{bmatrix}
\qquad
\vec{h} = \begin{bmatrix} 3 \\3 \\ 3 \\ 3 \end{bmatrix}
\end{align}


```
g = np.array([1, 1, 1, 8])
h = np.array([3, 3, 3, 3])

g_magnitude = np.sqrt((g**2).sum())
h_magnitude = np.sqrt((h**2).sum())

if g_magnitude > h_magnitude:
  print(f"The magnitude of g ({g_magnitude}) is greater than the magnitude of h ({h_magnitude})")
else:
    print(f"The magnitude of h ({h_magnitude}) is greater than the magnitude of g ({g_magnitude})")
```

    The magnitude of g (8.18535277187245) is greater than the magnitude of h (6.0)


# Part 2 - Matrices

## 2.1 What are the dimensions of the following matrices? Which of the following can be multiplied together? See if you can find all of the different legal combinations.
\begin{align}
A = \begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6
\end{bmatrix}
\qquad
B = \begin{bmatrix}
2 & 4 & 6 \\
\end{bmatrix}
\qquad
C = \begin{bmatrix}
9 & 6 & 3 \\
4 & 7 & 11
\end{bmatrix}
\qquad
D = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\qquad
E = \begin{bmatrix}
1 & 3 \\
5 & 7
\end{bmatrix}
\end{align}


```
A = np.array([[1, 2],
              [3, 4],
              [5,6]])
B = np.array([2, 4, 6])
C = np.array([[9, 6, 3],
              [4, 7, 11]])
D = np.array([[1, 0, 0],
              [0, 1, 0],
              [0, 0, 1]])
E = np.array([[1, 3],
              [5, 7]])
```

A - 3x2
> A*B = No // A*C = Yes // A*D = No // A*E = Yes

B - 1x3


> BA = Yes // BC = No // BD = Yes // BE = No


C - 2x3


> CA = Yes // CB = No // CD = Yes // CE = No

D - 3x3

> DA = Yes // DB = No // DC = No // DE = No

E - 2x2


> EA = No // EB = No // EC = Yes // ED = No



## 2.2 Find the following products: CD, AE, and BA. What are the dimensions of the resulting matrices? How does that relate to the dimensions of their factor matrices?


```
print(np.matmul(C,D))
print(np.matmul(A,E))
print(np.matmul(B,A))

```

    [[ 9  6  3]
     [ 4  7 11]]
    [[11 17]
     [23 37]
     [35 57]]
    [44 56]


CD = 2x3
AE = 3x2
BA = 1x2

## 2.3  Find $F^{T}$. How are the numbers along the main diagonal (top left to bottom right) of the original matrix and its transpose related? What are the dimensions of $F$? What are the dimensions of $F^{T}$?

\begin{align}
F = 
\begin{bmatrix}
20 & 19 & 18 & 17 \\
16 & 15 & 14 & 13 \\
12 & 11 & 10 & 9 \\
8 & 7 & 6 & 5 \\
4 & 3 & 2 & 1
\end{bmatrix}
\end{align}

The main diagonal in F, and it's transpose, F.T are the same.

F - 5x4
F.T - 4x5


```
F = np.array([[20,19,18,17],
              [16,15,14,13],
              [12,11,10,9],
              [8,7,6,5],
              [4,3,2,1]])

F.T
```




    array([[20, 16, 12,  8,  4],
           [19, 15, 11,  7,  3],
           [18, 14, 10,  6,  2],
           [17, 13,  9,  5,  1]])



# Part 3 - Square Matrices

## 3.1 Find $IG$ (be sure to show your work) 😃

You don't have to do anything crazy complicated here to show your work, just create the G matrix as specified below, and a corresponding 2x2 Identity matrix and then multiply them together to show the result. You don't need to write LaTeX or anything like that (unless you want to).

\begin{align}
G= 
\begin{bmatrix}
13 & 14 \\
21 & 12 
\end{bmatrix}
\end{align}


```
G = np.array([[13,14],
              [21,12]])

G_identity = np.array([[1,0],
                       [0,1]])

# If the product is equal to G, we have the proper IG
np.matmul(G, G_identity)
```




    array([[13, 14],
           [21, 12]])



## 3.2 Find $|H|$ and then find $|J|$.

\begin{align}
H= 
\begin{bmatrix}
12 & 11 \\
7 & 10 
\end{bmatrix}
\qquad
J= 
\begin{bmatrix}
0 & 1 & 2 \\
7 & 10 & 4 \\
3 & 2 & 0
\end{bmatrix}
\end{align}



```
H = np.array([[12,11],
              [7,10]])

J = np.array([[0,1,2],
              [7,10,4],
              [3,2,0]])
```


```
np.linalg.det(H)
```




    43.000000000000014




```
np.linalg.det(J)
```




    -19.999999999999996



## 3.3 Find $H^{-1}$ and then find $J^{-1}$

$H^{-1}$ = \begin{bmatrix}
10 & -11 \\
-7 & 12 
\end{bmatrix}



---


 $J^{-1}$ = No Determinant

## 3.4 Find $HH^{-1}$ and then find $J^{-1}J$. Is $HH^{-1} == J^{-1}J$? Why or Why not? 

Please ignore Python rounding errors. If necessary, format your output so that it rounds to 5 significant digits (the fifth decimal place).

I would say no, as $J^{-1}$ doesn't exist

# Stretch Goals: 

A reminder that these challenges are optional. If you finish your work quickly we welcome you to work on them. If there are other activities that you feel like will help your understanding of the above topics more, feel free to work on that. Topics from the Stretch Goals sections will never end up on Sprint Challenges. You don't have to do these in order, you don't have to do all of them. 

- Write a function that can calculate the dot product of any two vectors of equal length that are passed to it.
- Write a function that can calculate the norm of any vector
- Prove to yourself again that the vectors in 1.9 are orthogonal by graphing them. 
- Research how to plot a 3d graph with animations so that you can make the graph rotate (this will be easier in a local notebook than in google colab)
- Create and plot a matrix on a 2d graph.
- Create and plot a matrix on a 3d graph.
- Plot two vectors that are not collinear on a 2d graph. Calculate the determinant of the 2x2 matrix that these vectors form. How does this determinant relate to the graphical interpretation of the vectors?



73962d1a82ce5bebe6ae6304259cb30bfb2ad5be
ipynb
Jupyter Notebook
module1-vectors-and-matrices/Shawn_Harrington_LS_DS20_131_Vectors_and_Matrices_Assignment.ipynb
Shawn2776/Build_sample
d783717537b8618906f71a3306d918b571fe5f04
MIT
module1-vectors-and-matrices/Shawn_Harrington_LS_DS20_131_Vectors_and_Matrices_Assignment.ipynb
Shawn2776/Build_sample
d783717537b8618906f71a3306d918b571fe5f04
MIT
module1-vectors-and-matrices/Shawn_Harrington_LS_DS20_131_Vectors_and_Matrices_Assignment.ipynb
Shawn2776/Build_sample
d783717537b8618906f71a3306d918b571fe5f04
MIT
Qwen/Qwen-72B
1. YES
2. YES