After building up the $n$-chain groups $C_{n}(X)$ corresponding to our simplicial complex $X$, we defined two special groups: $B_{n}(X)$ and $Z_{n}(X)$, the $n$-boundaries and the $n$-cycles. These latter two groups will have somewhat of a visual interpretation (at least for small $n$) and will direct us into seeing, loosely, how many holes we can find in our structure. Moreover, for the structures we will be looking at, these calculations can be done by a computer! To do this, we'll have to bring up a bit of linear algebra.

What cycles and boundaries look like.

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