28.3: Practice Example

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

Consider the linear transformation defined by the following matrix \(A\).

\[\begin{split}A =
\left[
\begin{matrix}
1 & 2 & 3 & 1 \\
1 & 1 & 2 & 1 \\
1 & 2 & 3 & 1
\end{matrix}
\right]
\end{split} \nonumber \]

Question

What is the reduced row echelon form of \(A\)? You can use sympy.

#Put your answer to the above question here.

Question

Now let’s calculate the row space of \(A\). Note that the row space is defined by a linear combination of the non-zero row vectors in the reduced row echelon matrix:

Question

What is the rank of matrix \(A\)? You should know the rank by inspecting the reduced row echelon form. Find a numpy or sympy function that you can use to verify your answer?

## Put code here to verify your answer.

Question

Using the reduced row echelon form define the leading variables in terms of the free variables for the homogeneous equation.

Question

The solution to the above question defines the nullspace of \(A\) (aka the Kernel). Use the sympy.nullspace function to verify your answer.

# Put your code here

Question

Now let’s calculate the range of \(A\) (column space of \(A\)). Note that the range is spanned by the column vectors of \(A\). Transpose \(A\) and calculate the reduced row echelon form of the transposed matrix like we did above.

## Put your code here

Question

The nonzero row vectors of the above solution will give a basis for the range (or \(\mathcal{C}(A)\)). Write the range of \(A\) as a linear combination of these nonzero vectors:

Question

Finally, using the reduced row echelon form for \(A^{\top}\) define the leading variables in terms of the free variables and define the null space of \(A^{\top}\).