Title: Theory / Unimportant / Math / Lie / Naive theory

Name: The rotation of the plane \(\mathbb R^2\) about the origin through angle \(\theta\)
Definition: [$$]\left(\begin{array}{ll}\cos\theta&-\sin\theta\\ \sin\theta &\cos\theta\end{array}\right)[/$$]
Page: 2
Chapter: 1
Variables: \(\theta\in\mathbb R\)
Notation: \(R_\theta\)
Kind: Prop

Name: The special orthogonal group
Definition: The set of \(R_\theta\) for \(\theta\in\mathbb R\) under matrix composition
Page: 3
Chapter: 1
Notation: \(SO(2)\)
Kind: Prop

Name: A matrix Lie group
Definition: A set of \(n\times n\) matrices for composition, for some fixed \(n\), which is closed under nonsingular limit
Page: 4
Chapter: 1
Kind: Prop

Name: The nonsingular limit of \((A_i)_{i\in\mathbb N}\)
Definition: the limit of \((A_i)_{i\in\mathbb N}\), assuming it is a nonsingular matrices
Page: 4
Chapter: 1
Variables: \(n\in\mathbb N\)
\((A_i)_{i\in\mathbb N}\) \(n\times n\)-matrices
Kind: Prop

Name: The matrix representation of \(a+ib\)
Definition: [$$]\left(\begin{array}{ll}a&-b\\ b &a\end{array}\right)[/$$]
Page: 5
Chapter: 1
Section: 2
Variables: \(a+ib\in\mathbb C\)
Kind: Prop

Name: The matrix representation of \(a+ib+jc+kd\)
Definition: [$$]\left(\begin{array}{ll}a+id&-b-ic\\ b-ic &a-id\end{array}\right)[/$$]
Page: 7
Chapter: 1
Section: 3
Variables: \(a, b, c, d\in\mathbb R\)
Context: Quaternion
Kind: Prop

Name: The matrix representation of \(i\) as a quaternion
Definition: [$$]\left(\begin{array}{ll}0&-1\\1&0\end{array}\right)[/$$]
Page: 7
Chapter: 1
Section: 3
Context: Quaternion
Kind: Prop

Name: The matrix representation of \(j\) as a quaternion
Definition: [$$]\left(\begin{array}{ll}0&-i\\-i&0\end{array}\right)[/$$]
Page: 7
Chapter: 1
Section: 3
Context: Quaternion
Kind: Prop

Name: The matrix representation of \(k\) as a quaternion
Definition: [$$]\left(\begin{array}{ll}i&0\\0&-i\end{array}\right)[/$$]
Page: 7
Chapter: 1
Section: 3
Context: Quaternion
Kind: Prop

Name: A unit quaternion
Definition: A quaternion of norm 1
Page: 10
Chapter: 1
Section: 3

Name: A pure imaginary quaternion
Definition: A quaternion of the form \(bi+cj+dk\)
Page: 12
Chapter: 1
Section: 3

Name: The vector product of \(u\) and \(v\)
Definition: \(\left|\begin{array}{lll}i&j&k\\u_i&u_j&u_k\\v_i&v_j&v_k\end{array}\right|\)
Page: 13
Chapter: 1
Section: 4
Variables: \(u=u_ii+u_jj+u_kk\)
\(v=v_ii+v_jj+v_kk\)
Notation: \(u\times v\)
Definition2: \((u_jv_k-u_kv_j)i+(u_kv_i-u_iv_k)j+(u_iv_j-u_jv_i)k\)
Definition3: The vector, orthogonal to \(u\) and \(v\), whose length is the area spanned by \(u\) and \(v\), such that \(\det[u, v, u\times v]\ge 0\)
Name2: The cross product of \(u\) and \(v\)

Question: \(-u\times v\)
Answer: equals \(v\times u\)
Page: 13
Chapter: 1
Section: 4
Variables: \(u=u_ii+u_jj+u_kk\)
\(v=v_ii+v_jj+v_kk\)

Smaller0: \(|u|=1\)
Greater0: \(u^2=-1\)
Page: 14
Chapter: 1
Section: 5
Variables: \(u=u_ii+u_jj+u_kk\in\mathbb Ri+\mathbb Rj+\mathbb Rk\)
Context: Pure quaternion

Name: Antipodal quaternion
Definition: two quaternions \(u\) and \(-u\)
Page: 15
Chapter: 1
Section: 5

Name: the exceptional Lie group
Definition: the Lie group associated with octonions
Page: 22
Chapter: 1
Section: 5