17.3: One interpretation of determinants

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The following is an application of determinants. Watch this!

from IPython.display import YouTubeVideo
YouTubeVideo("Ip3X9LOh2dk",width=640,height=360, cc_load_policy=True)

For fun, we will recreate some of the video’s visualizations in Python. It was a little tricky to get the aspect ratios correct but here is some code I managed to get it work.

%matplotlib inline
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d.art3d import Poly3DCollection, Line3DCollection
import numpy as np
import sympy as sym

# Lets define somme points that form a Unit Cube
points = np.array([[0, 0, 0],
                  [1, 0, 0 ],
                  [1, 1, 0],
                  [0, 1, 0],
                  [0, 0, 1],
                  [1, 0, 1 ],
                  [1, 1, 1],
                  [0, 1, 1]])

points = np.matrix(points)

#Here is some code to build cube from https://stackoverflow.com/questions/44881885/python-draw-3d-cube

def plot3dcube(Z):
    
    if type(Z) == np.matrix:
        Z = np.asarray(Z)

    fig = plt.figure()
    ax = fig.add_subplot(111, projection='3d')

    r = [-1,1]

    X, Y = np.meshgrid(r, r)
    # plot vertices
    ax.scatter3D(Z[:, 0], Z[:, 1], Z[:, 2])

    # list of sides' polygons of figure
    verts = [[Z[0],Z[1],Z[2],Z[3]],
     [Z[4],Z[5],Z[6],Z[7]], 
     [Z[0],Z[1],Z[5],Z[4]], 
     [Z[2],Z[3],Z[7],Z[6]], 
     [Z[1],Z[2],Z[6],Z[5]],
     [Z[4],Z[7],Z[3],Z[0]], 
     [Z[2],Z[3],Z[7],Z[6]]]

    #alpha transparency was't working found fix here: 
    # https://stackoverflow.com/questions/23403293/3d-surface-not-transparent-inspite-of-setting-alpha
    # plot sides
    ax.add_collection3d(Poly3DCollection(verts, 
     facecolors=(0,0,1,0.25), linewidths=1, edgecolors='r'))
    
    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.set_zlabel('Z')
    
    ## Weird trick to get the axpect ratio to work.
    ## From https://stackoverflow.com/questions/13685386/matplotlib-equal-unit-length-with-equal-aspect-ratio-z-axis-is-not-equal-to
    mx = np.amax(Z, axis=0)
    mn = np.amin(Z, axis=0)
    max_range = mx-mn

    # Create cubic bounding box to simulate equal aspect ratio
    Xb = 0.5*max_range.max()*np.mgrid[-1:2:2,-1:2:2,-1:2:2][0].flatten() + 0.5*(max_range[0])
    Yb = 0.5*max_range.max()*np.mgrid[-1:2:2,-1:2:2,-1:2:2][1].flatten() + 0.5*(max_range[1])
    Zb = 0.5*max_range.max()*np.mgrid[-1:2:2,-1:2:2,-1:2:2][2].flatten() + 0.5*(max_range[2])
    # Comment or uncomment following both lines to test the fake bounding box:
    for xb, yb, zb in zip(Xb, Yb, Zb):
        ax.plot([xb], [yb], [zb], 'w')

    plt.show()

plot3dcube(points)

Question

The following the \(3 \times 3\) was shown in the video (around 6’50’’). Apply this matrix to the unit cube and use the plot3dcube to show the resulting transformed points.

T = np.matrix([[1 , 0 ,  0.5],
               [0.5 ,1 ,1.5],
               [1 , 0 ,  1]])

#Put the answer to the above question here.

Question

The determinant represents how the area changes when applying a \(2 \times 2 \) transform. What does the determinant represent for a \(3 \times 3\) transform?