21.3: Lots of Things Can Be Vector Spaces

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from IPython.display import YouTubeVideo
YouTubeVideo("YmGWj9RrNMI",width=640,height=360, cc_load_policy=True)

Consider the following two matrices \(A \in R^{3 \times 3}\) and \( B \in R^{3 \times 3} \), which consist of real numbers:

%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
sym.init_printing()

a11,a12,a13,a21,a22,a23,a31,a32,a33 = sym.symbols('a_{11},a_{12}, a_{13},a_{21},a_{22},a_{23},a_{31},a_{32},a_{33}', negative=False)
A = sym.Matrix([[a11,a12,a13],[a21,a22,a23],[a31,a32,a33]])
A

b11,b12,b13,b21,b22,b23,b31,b32,b33 = sym.symbols('b_{11},b_{12}, b_{13},b_{21},b_{22},b_{23},b_{31},b_{32},b_{33}', negative=False)
B = sym.Matrix([[b11,b12,b13],[b21,b22,b23],[b31,b32,b33]])
B

Question

What properties do we need to show all \(3 \times 3\) matrices of real numbers form a vector space.

Do This

Demonstrate these properties using sympy as was done in the video.

#Put your answer here.

Question (assignment specific)

Determine whether \(A\) is a linear combination of \(B\), \(C\), and \(D\)?

\[\begin{split} A=
\left[
\begin{matrix}
7 & 6 \\
-5 & -3
\end{matrix}
\right],
B=
\left[
\begin{matrix}
3 & 0 \\
1 & 1
\end{matrix}
\right],
C=
\left[
\begin{matrix}
0 & 1 \\
3 & 4
\end{matrix}
\right],
D=
\left[
\begin{matrix}
1 & 2 \\
0 & 1
\end{matrix}
\right]
\end{split} \nonumber \]

#Put your answer to the above question here

Question

Write a basis for all \(2 \times 3\) matrices and give the dimension of the space.