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A2569875/模板沙盒/數學測試/3
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<
User:A2569875
|
模板沙盒
|
數學測試
[
cos
(
45
∘
)
−
sin
(
45
∘
)
sin
(
45
∘
)
cos
(
45
∘
)
]
×
v
e
c
t
o
r
(
1
2
,
1
2
)
=
[
0
1
]
{\displaystyle {{\begin{bmatrix}{\cos \left({{45}\,{{}^{\circ }}}\right)}&{-{\sin \left({{45}\,{{}^{\circ }}}\right)}}\\{\sin \left({{45}\,{{}^{\circ }}}\right)}&{\cos \left({{45}\,{{}^{\circ }}}\right)}\end{bmatrix}}\times {vector\left({\frac {1}{\sqrt {2}}},\,{\frac {1}{\sqrt {2}}}\right)}}={\begin{bmatrix}{0}\\{1}\end{bmatrix}}}
[
cos
(
45
∘
)
−
sin
(
45
∘
)
sin
(
45
∘
)
cos
(
45
∘
)
]
[
1
0
]
=
[
0.70710678118655
0.70710678118655
]
{\displaystyle {{\begin{bmatrix}{\cos \left({{45}\,{{}^{\circ }}}\right)}&{-{\sin \left({{45}\,{{}^{\circ }}}\right)}}\\{\sin \left({{45}\,{{}^{\circ }}}\right)}&{\cos \left({{45}\,{{}^{\circ }}}\right)}\end{bmatrix}}\,{\begin{bmatrix}1\\0\end{bmatrix}}}={\begin{bmatrix}{0.70710678118655}\\{0.70710678118655}\end{bmatrix}}}
[
cos
(
60
∘
)
−
sin
(
60
∘
)
sin
(
60
∘
)
cos
(
60
∘
)
]
[
3
2
1
2
]
=
[
0
1
]
{\displaystyle {{\begin{bmatrix}{\cos \left({{60}\,{{}^{\circ }}}\right)}&{-{\sin \left({{60}\,{{}^{\circ }}}\right)}}\\{\sin \left({{60}\,{{}^{\circ }}}\right)}&{\cos \left({{60}\,{{}^{\circ }}}\right)}\end{bmatrix}}\,{\begin{bmatrix}{\frac {\sqrt {3}}{2}}\\{\frac {1}{2}}\end{bmatrix}}}={\begin{bmatrix}{0}\\{1}\end{bmatrix}}}
round
(
log
2
[
1
2
3
5
7
6
7
9
8
]
,
3
)
=
[
0.398
+
3.532
i
1.483
+
1.53
i
−
0.074
−
2.166
i
1.409
−
0.637
i
0.52
−
0.276
i
2.286
+
0.39
i
1.36
−
2.081
i
2.663
−
0.901
i
1.667
+
1.276
i
]
{\displaystyle \operatorname {round} \left(\log _{2}{\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,3\right)={\begin{bmatrix}{0.398+3.532i}&{1.483+1.53i}&{-0.074-2.166i}\\{1.409-0.637i}&{0.52-0.276i}&{2.286+0.39i}\\{1.36-2.081i}&{2.663-0.901i}&{1.667+1.276i}\end{bmatrix}}}
round
(
2
log
2
[
1
2
3
5
7
6
7
9
8
]
,
3
)
=
[
1
2
3
5
7
6
7
9
8
]
{\displaystyle \operatorname {round} \left({{2}^{\log _{2}{\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}}},\,3\right)={\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}}
round
(
ln
[
1
2
3
5
7
6
7
9
8
]
,
3
)
=
[
0.276
+
2.448
i
1.028
+
1.06
i
−
0.051
−
1.501
i
0.977
−
0.441
i
0.361
−
0.191
i
1.585
+
0.271
i
0.943
−
1.442
i
1.846
−
0.625
i
1.156
+
0.885
i
]
{\displaystyle \operatorname {round} \left(\ln {\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}},\,3\right)={\begin{bmatrix}{0.276+2.448i}&{1.028+1.06i}&{-0.051-1.501i}\\{0.977-0.441i}&{0.361-0.191i}&{1.585+0.271i}\\{0.943-1.442i}&{1.846-0.625i}&{1.156+0.885i}\end{bmatrix}}}
round
(
e
ln
[
1
2
3
5
7
6
7
9
8
]
,
3
)
=
[
1
2
3
5
7
6
7
9
8
]
{\displaystyle \operatorname {round} \left(e^{\ln {\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{1}&{2}&{3}\\{5}&{7}&{6}\\{7}&{9}&{8}\end{bmatrix}}}
round
(
ln
[
1
2
5
7
]
,
3
)
=
[
−
0.534
+
2.652
i
0.722
−
0.721
i
1.806
−
1.802
i
1.633
+
0.49
i
]
{\displaystyle \operatorname {round} \left(\ln {\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}},\,3\right)={\begin{bmatrix}{-0.534+2.652i}&{0.722-0.721i}\\{1.806-1.802i}&{1.633+0.49i}\end{bmatrix}}}
round
(
e
ln
[
1
2
5
7
]
,
3
)
=
[
1
2
5
7
]
{\displaystyle \operatorname {round} \left(e^{\ln {\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}}
round
(
e
ln
[
1
2
5
7
]
×
[
3
6
7
9
]
,
3
)
=
[
111880867330.3
+
289269930190.11
i
160816129651.98
+
418718548369.82
i
753700363714.63
−
111717403084.54
i
1090370340897.3
−
158999398148.36
i
]
{\displaystyle \operatorname {round} \left(e^{{\ln {\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}}\times {\begin{bmatrix}{3}&{6}\\{7}&{9}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{111880867330.3+289269930190.11i}&{160816129651.98+418718548369.82i}\\{753700363714.63-111717403084.54i}&{1090370340897.3-158999398148.36i}\end{bmatrix}}}
round
(
[
1
2
5
7
]
[
3
0
0
3
]
,
3
)
=
[
91
134
335
493
]
{\displaystyle \operatorname {round} \left({{\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}^{\begin{bmatrix}{3}&{0}\\{0}&{3}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{91}&{134}\\{335}&{493}\end{bmatrix}}}
round
(
log
[
1
2
5
7
]
(
[
1
2
5
7
]
[
3
0
0
3
]
)
,
3
)
=
[
1.474
+
0.498
i
0.415
−
0.135
i
1.037
−
0.338
i
2.718
+
0.092
i
]
{\displaystyle \operatorname {round} \left(\log _{\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}\left({{\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}^{\begin{bmatrix}{3}&{0}\\{0}&{3}\end{bmatrix}}}\right),\,3\right)={\begin{bmatrix}{1.474+0.498i}&{0.415-0.135i}\\{1.037-0.338i}&{2.718+0.092i}\end{bmatrix}}}
round
(
[
2
0
0
2
]
[
3
0
0
3
]
,
3
)
=
[
8
0
0
8
]
{\displaystyle \operatorname {round} \left({{\begin{bmatrix}{2}&{0}\\{0}&{2}\end{bmatrix}}^{\begin{bmatrix}{3}&{0}\\{0}&{3}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{8}&{0}\\{0}&{8}\end{bmatrix}}}
round
(
log
[
2
0
0
2
]
(
[
2
0
0
2
]
[
3
0
0
3
]
)
,
3
)
=
[
3
0
0
3
]
{\displaystyle \operatorname {round} \left(\log _{\begin{bmatrix}{2}&{0}\\{0}&{2}\end{bmatrix}}\left({{\begin{bmatrix}{2}&{0}\\{0}&{2}\end{bmatrix}}^{\begin{bmatrix}{3}&{0}\\{0}&{3}\end{bmatrix}}}\right),\,3\right)={\begin{bmatrix}{3}&{0}\\{0}&{3}\end{bmatrix}}}
round
(
[
1
2
5
7
]
[
3
6
7
9
]
,
3
)
=
[
111880867330.23
+
289269930190.14
i
160816129651.89
+
418718548369.86
i
753700363714.65
−
111717403084.35
i
1090370340897.3
−
158999398148.09
i
]
{\displaystyle \operatorname {round} \left({{\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}^{\begin{bmatrix}{3}&{6}\\{7}&{9}\end{bmatrix}}},\,3\right)={\begin{bmatrix}{111880867330.23+289269930190.14i}&{160816129651.89+418718548369.86i}\\{753700363714.65-111717403084.35i}&{1090370340897.3-158999398148.09i}\end{bmatrix}}}
round
(
log
[
1
2
5
7
]
(
[
1
2
5
7
]
[
3
6
7
9
]
)
,
3
)
=
[
2.384
+
2.358
i
2.506
+
4.378
i
7.237
−
1.673
i
11.409
−
2.956
i
]
{\displaystyle \operatorname {round} \left(\log _{\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}\left({{\begin{bmatrix}{1}&{2}\\{5}&{7}\end{bmatrix}}^{\begin{bmatrix}{3}&{6}\\{7}&{9}\end{bmatrix}}}\right),\,3\right)={\begin{bmatrix}{2.384+2.358i}&{2.506+4.378i}\\{7.237-1.673i}&{11.409-2.956i}\end{bmatrix}}}
