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title: TDA_TEAM_4

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title: TDA_TEAM_4
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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.
$$0\quad \rightarrow\quad C_1\quad \rightarrow\quad C_0\quad \rightarrow\quad 0$$

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 image of del2 has to be zero because it is a linear transformation.



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

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.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0\\1&0&1&0\\ 0&1&1&1\\0&0&0&1\end{bmatrix}\sim \begin{bmatrix}1&0&1&0\\0&1&1&0\\ 0&0&0&0\\0&0&0&1\end{bmatrix}$$

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 image of del2 has to be zero because it is a linear transformation.




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

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.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0&0\\1&0&1&1&0\\ 0&1&1&0&1\\0&0&0&1&1\end{bmatrix}\sim \begin{bmatrix}1&0&1&0&1\\0&1&1&0&1\\ 0&0&0&1&1\\0&0&0&0&0\end{bmatrix}$$

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

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.
The image of del3 has to be zero because it is a linear transformation.


dim$H_0$ tells us how many subsets we can break the verticies into, based on which ones are connected.

For example A dim$H_0=2$ and the subsets are $\{0,1,2\}$ and $\{3\}$.

For example B and C dim$H_0=1$ and the verticies belong to the same set since they are all connected

dim$H_1$ and dim$H_2$ tells us how many subsets we can break the edges and areas into, respectively.