京都大学 情報学研究科 知能情報学専攻 2022年8月実施 情報学基礎 F1-1
Author
Description
Kai
設問1
\[
A=
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
-7 & 1 & 0 & 0 & 0 \\
-5 & 2 & 1 & 0 & 0 \\
7 & -4 & -4 & 1 & 0 \\
-2 & 0 & -4 & -9 & 1
\end{bmatrix}
\begin{bmatrix}
-6 & -9 & -2 & 7 & -9 \\
0 & -4 & -7 & -4 & 7 \\
0 & 0 & -4 & 8 & -1 \\
0 & 0 & 0 & 2 & -2 \\
0 & 0 & 0 & 0 & 3
\end{bmatrix}
\]
設問2
(1)
\[
Q\bar{Q} = (a^2+b^2+c^2+d^2)E
\]
(2)
\[
I^2 = -E \Rightarrow I^{-1} =
\begin{bmatrix}
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0 \\
\end{bmatrix}
\]
\[
Q\bar{Q} = (a^2+b^2+c^2+d^2)E \Rightarrow Q^{-1} = \frac{1}{a^2+b^2+c^2+d^2}
\begin{bmatrix}
a & -b & -c & d \\
b & a & d & -c \\
c & -d & a & b \\
d & c & -b & a \\
\end{bmatrix}
\]
(3)
The complete proving is to use \((a, b, c, d)\) to represent all the \(Q\)s below with their calculations, which is easy but tedious.
- (a)
\[
\forall Q_1, Q_2 \in H, Q_1 + Q_2 \in H
\]
- (b)
\[
\forall Q_1, Q_2 \in H, Q_1 + Q_2 = Q_2 + Q_1
\]
- (c)
\[
\forall Q_1, Q_2, Q_3 \in H, (Q_1 + Q_2) + Q_3 = Q_1 + (Q_2 + Q_3)
\]
- (d)
\[
\exists Q_0 \in H, \forall Q \in H, Q_0 + Q = Q
\]
- (e)
\[
\forall Q \in H, \exists \hat{Q} \in H, Q + \hat{Q} = Q_0
\]
- (f)
\[
\forall Q_1, Q_2 \in H, Q_1Q_2 \in H
\]
- (g)
\[
\forall Q_1, Q_2, Q_3 \in H, (Q_1Q_2)Q_3 = Q_1(Q_2Q_3)
\]
- (h)
\[
\forall Q_1, Q_2, Q_3 \in H, (Q_1 + Q_2)Q_3 = Q_1Q_3 + Q_2Q_3
\]
- (i)
\[
\exists Q_1, Q_2 \in H, Q_1Q_2 \neq Q_2Q_1
\]