Extra DQ 1 - CoCalc

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Description: Example showing two interpretations of Mx = b.

Compute Environment: Ubuntu 20.04 (Default)

In [2]:

x,y,z,t = var('x,y,z,t')
M = matrix([[1,2,3],[2,-3,1],[1,1,0]])
v = matrix([x,y,z]).transpose()
F = M*v
p = matrix([1,1,1]).transpose()
b = M*p

This first example shows the interpretation of the system $M\mathbf{x}=\mathbf{b}$ as the intersection of planes where $\begin{bmatrix}1&2&3\\2&-3&1\\1&1&0\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}6\\0\\2\end{bmatrix}$ is viewed as a system of equations $\begin{alignat}{7} & &&x &&+ 2&&y &&+3&&z &&=6\\ &2&&x &&- 3&&y &&+ &&z &&=0\\ & &&x &&+ &&y && && &&= 2 \end{alignat}$ The solution here is $x=1, y=1, z=1$

In [3]:

G = implicit_plot3d(F[0][0] == 6, (x,-4,4), (y, -4,4), (z, -4, 4), color='green', opacity='0.9', aspect_ratio=[1,1,1])
G += implicit_plot3d(F[1][0] == 0, (x,-4,4), (y, -4,4), (z, -4, 4), color='red', opacity='0.9')
G += implicit_plot3d(F[2][0] == 2, (x,-4,4), (y, -4,4), (z, -4, 4), color='blue', opacity='0.9')
G += sphere((1,1,1), size=.1, color='yellow', opacity='1')
G.show(fig_size=4)