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title: TDA_team_1
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**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.
$$0\quad \rightarrow\quad C_1\quad \rightarrow\quad C_0\quad \rightarrow\quad 0$$

kernel of $\partial_2$=0
image of $\partial_2$=0

kernel of $\partial_0=C_0=$span{0,1,2,3}=4
image of $\partial_0$=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.

The dimension of $H_0$ is describing the number of clusters in our space, in this case there are two. The dimension of $H_1$ describes the number of voids that we have in our space, in this case there is one.

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

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

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

kernel of $\partial_2$=0
image of $\partial_2$=0

kernel of $\partial_0=C_0=$span{0,1,2,3}
image of $\partial_0$=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.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0\\1&0&1&0\\ 0&1&1&1\\0&0&0&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0\\0&1&1&0\\ 0&0&0&1\\0&0&0&0\end{bmatrix}$$

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.

The dimension of $H_0$ is describing the number of clusters in our space, in this case there is one. The dimension of $H_1$ describes the number of voids that we have in our space, in this case there is one.

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

kernel of $\partial_2$=0
image of $\partial_2$=0

kernel of $\partial_0=C_0=$span{0,1,2,3}
image of $\partial_0$=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.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0&0\\1&0&1&1&0\\ 0&1&1&0&1\\0&0&0&1&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0&0\\0&1&1&1&0\\ 0&0&0&1&1\\0&0&0&0&0\end{bmatrix}$$

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

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

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

The dimension of $H_0$ is describing the number of clusters in our space, in this case there is one. The dimension of $H_1$ describes the number of voids that we have in our space, in this case there are two.