---
title: TDA_team_3
---
==perfect==
**Homologies: $H_i=ker(\partial_i)/im(\partial_{i+1})$**

$$\dim H_i=\dim~ker(\partial_i)-\dim~im(\partial_{i+1})$$

**Example A.**
1. Create the chain.
$$0\quad \rightarrow\quad C_1\quad \rightarrow\quad C_0\quad \rightarrow\quad 0$$

$$\partial_2:0\rightarrow C_1$$

$ker(\partial_2)=0$, image$(\partial_2)=0$ because $0(01)+0(02)+0(12)=0$

$$\partial_0: C_2\rightarrow 0$$

$ker(\partial_0)= C_0=Span\{0,1,2,3\}$, image$(\partial_0)=0$

2. Describe the simplicial complexes

$$C_0:=\{0,1,2,3\},\quad C_1:=\{01,02,12\}$$

3. Create matrices to describe boundary maps.
$$\partial_1:\quad \begin{bmatrix}1&1&0\\1&0&1\\ 0&1&1\\0&0&0\end{bmatrix}\sim \begin{bmatrix}1&0&1\\0&1&1\\ 0&0&0\\0&0&0\end{bmatrix}$$

4. Compute homologies.
$$\dim H_0=\dim~ker(\partial_0)-\dim~im(\partial_{1})=4-2=2$$

$$\dim H_1=\dim~ker(\partial_1)-\dim~im(\partial_{2})=1-0=1$$

5. Describe your findings in your own words. AS MUCH AS YOU CAN.

$\dim H_0=2$ (the number of clusters).

$\dim H_1=1$ (the number of closed loops).

**Example B.**
1. Create the chain.
$$0\quad \rightarrow\quad C_1\quad \rightarrow\quad C_0\quad \rightarrow\quad 0$$

$$\partial_2:0\rightarrow C_1$$

$ker(\partial_2)=0$, image$(\partial_2)=0$ because $0(01)+0(02)+0(12)+0(23)=0$

$$\partial_0: C_2\rightarrow 0$$

$ker(\partial_0)= C_0=Span\{0,1,2,3\}$, image$(\partial_0)=0$

2. Describe the simplicial complexes

$$C_0:=\{0,1,2,3\},\quad C_1:=\{01,02,12,23\}$$

3. Create matrices to describe boundary maps.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0\\1&0&1&0\\ 0&1&1&1\\0&0&0&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0\\0&1&1&0\\ 0&1&1&1\\0&0&0&1\end{bmatrix} \sim \begin{bmatrix}1&1&0&0\\0&1&1&0\\ 0&0&0&1\\0&0&0&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0\\0&1&1&0\\ 0&0&0&1\\0&0&0&0\end{bmatrix}$$

4. Compute homologies.
$$\dim H_0=\dim~ker(\partial_0)-\dim~im(\partial_{1})=4-3=1$$

$$\dim H_1=\dim~ker(\partial_1)-\dim~im(\partial_{2})=1-0=1$$

5. Describe your findings in your own words. AS MUCH AS YOU CAN.

$\dim H_0=1$ (the number of clusters).

$\dim H_1=1$ (the number of closed loops).

**Example C.**
1. Create the chain.
$$0\quad \rightarrow\quad C_2\quad \rightarrow\quad C_1\quad \rightarrow\quad C_0\quad \rightarrow0$$

$$\partial_3:0\rightarrow C_1$$

$ker(\partial_3)=0$, image$(\partial_3)=0$ because $0(012)=0$

$$\partial_0: C_0\rightarrow 0$$

$ker(\partial_0)= C_0=Span\{0,1,2,3\}$, image$(\partial_0)=0$

2. Describe the simplicial complexes

$$C_0:=\{0,1,2,3\},\quad C_1:=\{01,02,12,13,23\},\quad C_2:=\{012\}$$

3. Create matrices to describe boundary maps.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0&0\\1&0&1&1&0\\ 0&1&1&0&1\\0&0&0&1&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0&0\\0&1&1&1&0\\ 0&1&1&0&1\\0&0&0&1&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0&0\\0&1&1&1&0\\ 0&0&0&1&1\\0&0&0&1&1\end{bmatrix}\sim\begin{bmatrix}1&1&0&0&0\\0&1&1&1&0\\ 0&0&0&1&1\\0&0&0&0&0\end{bmatrix}$$

$$\partial_2:\quad \begin{bmatrix} 1\\1\\1\\0\\0\end{bmatrix}\sim\begin{bmatrix}1\\0\\0\\0\\0\end{bmatrix}$$

4. Compute homologies.
$$\dim H_0=\dim~ker(\partial_0)-\dim~im(\partial_{1})=4-3=1$$

$$\dim H_1=\dim~ker(\partial_1)-\dim~im(\partial_{2})=2-1=1$$

$$\dim H_2=\dim~ker(\partial_2)-\dim~im(\partial_{3})=0-0=0$$

5. Describe your findings in your own words. AS MUCH AS YOU CAN.

$\dim H_0=1$ (the number of clusters).

$\dim H_1=1$ (the number of closed loops).

$\dim H_2=0$ (the number of three dimensional voids).