Activity 2.6.1.
We want to look at the effect that a matrix has when we look at the matrix’s effect on an entire vector space. Let’s start small and look at
\begin{equation*}
A=\begin{bmatrix} 2 \amp 5\\ 5 \amp 2 \end{bmatrix}
\end{equation*}
We want to look at what happens to different directions (measured with unit vectors) when we use the function \(T_A (\vec{x}): \vec{x} \rightarrow A\vec{x}\text{.}\)
(a)
Use Python to construct a bunch of unit vectors centered at the origin. Plot these vectors.
(b)
Use Python to calculate \(T_A(\vec{x})\) for each of your unit vectors. Plot these vectors. What do you notice about these results?
(c)
Where does \(T_A\) send \(\colvec{1\\0}\text{?}\) Why should this make sense on your plot?
(d)
Where does \(T_A\) send \(\colvec{0\\1}\text{?}\) Why should this make sense on your plot?
(e)
How does the area traced out by all unit vectors starting at the origin change under \(T_A\text{?}\) Calculate the determinant of \(A\) and compare to your change in area.
(f)
Where does \(T_A\) send \(\colvec{\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}}\text{?}\) Why should this make sense on your plot?
(g)
Are there other directions that work like this?
(h)
So \(\vec{v}_1= \colvec{\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}}\) and \(\vec{v}_2=\colvec{\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}}\) are important directions. Will there be any other directions that are just scalar multiplication under the action of \(T_A\text{?}\)
(i)
The set \(\{\vec{v}_1,\vec{v}_2\}\) spans all of \(\mathbb{R}^2\text{.}\) How can we justify this statement?
(j)
We can write any vector in \(\mathbb{R}^2\) as a linear combination of the set \(\{\vec{v}_1,\vec{v}_2\}\text{.}\) Use this idea to describe \(T_A(\vec{w})\) in terms of \(\vec{v}_1\) and \(\vec{v}_2\text{.}\)
(k)
Can we do all of these steps for other matrices? Great question Dr. Long. You deserve a raise and come cookies. You may proceed to the next section to see the answer.
