Introduction

Student Conversations

• Representing two component vectors in pairs, where one person is the top component and the other is the bottom component. These simple examples help to orient the students to representing two component vectors in pairs. \[\pmatrix{1\\i} \rm{and} \pmatrix{1\\-i}\]
• Transformations of two component vectors. This example highlights the idea that what is meant by an eigenvalue is that both components have the same thing happen to them and that if the eigenvalue is complex, it means a rotation in the complex plane. \[e^{i\pi/4}\pmatrix{1\\i}=\pmatrix{e^{i\pi/4}\\ie^{i\pi/4}}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}\]
• This provides an example of a transformation where different things happen to each component. \[\pmatrix{i&0\\0&1}\pmatrix{1\\i}=\pmatrix{i\\i}\]
• This is an example of a rotation, a transformation where both components stay real and the total length stays constant. The relative magnitude of the two components depends on the value of \(\theta\). \[\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\0}=\pmatrix{\cos\theta\\-\sin\theta}\] \[\rightarrow\dfrac{1}{\sqrt{2}}\pmatrix{1\\-1}\] when \(\theta=\dfrac{\pi}{4}\)
• This next example is the same rotation, now acting on a two-component complex vector and can be used to highlight the connection between transformations and eigenvalues and eigenvectors. \[\pmatrix{\cos\theta&\sin\theta\\-\sin\theta&\cos\theta}\pmatrix{1\\i}=\pmatrix{\cos\theta+i\sin\theta\\-\sin\theta+i\cos\theta}=\dfrac{1}{\sqrt{2}}\pmatrix{1+i\\-1+i}=\pmatrix{e^{i\pi/4}\\e^{i3\pi/4}}=e^{i\pi/4}\pmatrix{1\\i}\]

Wrap-up