24.2: Matrix Representation of Vector Spaces

%matplotlib inline
import matplotlib.pylab as plt
import numpy as np
import sympy as sym
from urllib.request import urlretrieve
sym.init_printing(use_unicode=True)
urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

Consider the following matrix \(A\).

\[\begin{split}
\left[
\begin{matrix}
1 & 0 & 3 \\
0 & 1 & 5 \\
1 & 1 & 8
\end{matrix}
\right]
\end{split} \nonumber \]

Question

What is the reduced row echelon form of \(A\)?

# Put your answer to the above question here.

from answercheck import checkanswer

checkanswer.matrix(rref,'1731818a1555cc33a778a4eb76af945c');

ROW SPACE The first and second (non zero) rows of the above matrix “spans” the same space as the orignal three row vectors in \(A\). We often call this the “row space” and it can be written as a linear combination of the non-zero rows of the reduced row echelon form:

\[row(A) = r(1,0,3)^\top+s(0,1,5)^\top \nonumber \]

Do This

Calculate the solutions to the system of homogeneous equations \(Ax=0\). This is often called the NULL SPACE or sometimes KERNEL of \(A\).

#Put your answer here

Do This

We introduced two subspaces. Pick one vector from the row space of \(A\) and another vector from the null space of \(A\). Find the dot product of these two vector.

#Put your answer here

Question

Did you get the same value for the dot product? Explain your answer.

Do This

What is the reduced row echelon form of \(A^T\)?

#Put your answer here

COLUMN SPACE: The first and second (non zero) rows of the above matrix “spans” the same space as the original three column vectors in \(A\). We often call this the “column space” (or “image space”) of \(A\) and it can be written as a linear combination of the non-zero rows of the reduced row echelon form of \(A^T\):

\[col(A) = a(1,0,1)^\top+b(0,1,1)^\top \nonumber \]

Do This

Calculate the solutions to the system of homogeneous equations \(A^T x=0\). This is often called the NULL SPACE of \(A^T\).

#Put your answer here

Example #1

Consider the following system of linear equations.

\[ x_1 - x_2 + x_3 = 3 \nonumber \]

\[ -2x_1 + 2x_2 - 2x_3 = -6 \nonumber \]

Do This

What are the solutions to the above system of equations?

# Put your code here

Do This

Come up with a specific arbitrary solution (any solution will do) to the above set of equations.

Do This

Now consider only the left hand side of the above matrix and solve for the kernel (null Space) of A:

\[\begin{split} A =
\left[
\begin{matrix}
1 & -1 & 1 \\
-2 & 2 & -2
\end{matrix}
\right]
\end{split} \nonumber \]

#Put your answer here

Do This

Express an arbitrary solution as the sum of an element of the kernel of the transformation defined by the matrix of coefficients and a particular solution.

Do This

Discuss in your group and the class your solution from above. How does the solution to \(Ax=b\) relate to the solution to \(Ax=0\). If you were to plot all solutions, what shape does it take? How does this shape relate to the kernel?