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title: TDA_team_4
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**Homologies: $H_i=ker(\partial_i)/im(\partial_{i+1})$**

$5x-3y=2$
limit~~**~~
$$\dim H_i=\dim~ker(\partial_i)-\dim~im(\partial_{i+1})$$

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

$\partial_2:0\rightarrow\ C_1$
$ker\partial_2=0$ because this is the domain.
$im\partial_2=0$ because it is a linear transformation.

$\partial_0:C_0\rightarrow\ 0\\
ker\partial_0=C_0=span~(0, 1, 2, 3),~dim=4$
$im\partial_0=0$ because the transformation maps everything to 0, there are no other possibilities.

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

$\dim H_0$ represents the number of connected by edges components.

$\dim H_1$ represents the number of enclosed spaces.

**Example B**

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

$\partial_2:0\rightarrow\ C_1$
$ker\partial_2=0$ because this is the domain.
$im\partial_2=0$ because it is a linear transformation.

$\partial_1:C_1\rightarrow\ C_0$

$\partial_0:C_0\rightarrow\ 0\\
ker\partial_0=C_0=span~(0, 1, 2, 3),~dim=4$
$im\partial_0=0$ because the transformation maps everything to 0, there are no other possibilities.

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

$\dim H_0$ represents the number of connected by edges components, or clusters.

$\dim H_1$ represents the number of enclosed spaces.

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

$\partial_3:0 \rightarrow C_2$
$ker\partial_3=0$ because this is the domain.
$im\partial_3=0$ because it is a linear transformation.

$\partial_2:C_2\rightarrow\ C_1$

$\partial_1:C_1\rightarrow\ C_0$

$\partial_0:C_0\rightarrow\ 0\\
ker\partial_0=C_0=span~(0, 1, 2, 3),~dim=4$
$im\partial_0=0$ because the transformation maps everything to 0, there are no other possibilities.

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.
$$\partial_1:\quad \begin{bmatrix}1&1&0&0&0\\1&0&1&1&0\\ 0&1&1&0&1\\0&0&0&1&1\end{bmatrix}\sim \begin{bmatrix}1&1&0&0&0\\0&1&1&1&0\\ 0&0&0&1&1\\0&0&0&0&0\end{bmatrix}$$
$\dim~ker\partial_1=2$ because there are 2 free variables.
$\dim~im\partial_1=3$ because there are 3 pivots.
$$\partial_2:\quad \begin{bmatrix}1\\1\\1\\ 0\\0\end{bmatrix}\sim \begin{bmatrix}1\\0\\ 0\\0\\0\end{bmatrix}$$
$\dim~ker\partial_2=0$ because there are no free variables and 0 vector is the only thing in $ker\partial_2$.
$\dim~im\partial_2=1$ because there is 1 pivot. $im\partial_2$ span vector $<1,1,1,0,0>$
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

$\dim H_0$ represents the number of connected by edges components, or clusters.

$\dim H_1$ represents the number of enclosed spaces.

====$\dim H_2$ represents the number of 3-dimensional void====
should be $H_2$ not $H_3$