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mermaid
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kenu committed Jul 19, 2024
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Expand Up @@ -57,32 +57,45 @@ graph LR
```

- polar form
$$a+ib = r(cos\theta+i sin\theta)$$
```mermaid
graph LR
A["$$a+ib = r(cos\theta+i sin\theta)$$"]
```

- exponential form
$$a+ib=re^{i\theta}$$
```mermaid
graph LR
A["$$a+ib=re^{i\theta}$$"]
```

$$1+i
=\sqrt{2}(cos(\frac{\pi}{4})+i sin(\frac{\pi}{4}))
=\sqrt{2}e^{i\frac{\pi}{4}}$$
```mermaid
graph LR
B["$$1+i=\sqrt{2}(cos(\frac{\pi}{4})+i sin(\frac{\pi}{4}))=\sqrt{2}e^{i\frac{\pi}{4}}$$"]
```

$$2\sqrt{2}e^{i\frac{\pi}{4}}$$
$$2\sqrt{2}e^{i\frac{-\pi}{2}}$$
$$2\sqrt{2}e^{i\frac{3\pi}{4}}$$
```mermaid
graph LR
A["$$2\sqrt{2}e^{i\frac{\pi}{4}}$$"]
A["$$2\sqrt{2}e^{i\frac{-\pi}{2}}$$"]
A["$$2\sqrt{2}e^{i\frac{3\pi}{4}}$$"]
$$e^{i\frac{\pi}{5}}e^{i\frac{\pi}{3}} = e^{i\frac{8\pi}{15}}$$
A["$$e^{i\frac{\pi}{5}}e^{i\frac{\pi}{3}} = e^{i\frac{8\pi}{15}}$$"]
```

### Matricies

$$\begin{bmatrix}1\space -2\\0\space -10\end{bmatrix}$$
$$\begin{bmatrix}1\\2\\-3\\8\end{bmatrix}$$
$$\begin{bmatrix}5\space-2\space3\\7\space2\space1.5\end{bmatrix}$$
```mermaid
graph LR
A["$$\begin{bmatrix}1\space -2\\0\space -10\end{bmatrix}$$"]
A["$$\begin{bmatrix}1\\2\\-3\\8\end{bmatrix}$$"]
A["$$\begin{bmatrix}5\space-2\space3\\7\space2\space1.5\end{bmatrix}$$"]
```

#### m x n matrix
```mermaid
graph LR
A["$$\begin{bmatrix}1\space -2\space 3\\0\space -10\space 4\\1\space 2\space 3\end{bmatrix}$$"]
$$\begin{bmatrix}1\space -2\space 3\\0\space -10\space 4\\1\space 2\space 3\end{bmatrix}$$

---

$$\begin{pmatrix} 1 & 1 & 2 \\ -1 & 2 & 1 \ \end{pmatrix}$$
B["$$\begin{pmatrix} 1 & 1 & 2 \\ -1 & 2 & 1 \ \end{pmatrix}$$"]
$$\text{det} = 1(-1) - (-1)(2) = -1 + 2 = 1$$
C["$$\text{det} = 1(-1) - (-1)(2) = -1 + 2 = 1$$"]
```

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