## Key Sequence

- we defined subspace and how to check for them
- we want to operate on subsets, so we defined the sum of subsets
- we saw that the sum of subspaces are the smallest containing subspace
- and finally, we defined direct sums and how to prove them

## New Definitions

## Results and Their Proofs

- checking for subspaces
- creating direct sums

## Questions for Jana

~~Does the additive identity have be the same between different subspaces of the same vector space?~~yes, otherwise the larger vector space has two additive identities.~~Does the addition and multiplication operations in a subspace have to be the same as its constituent vector space?~~by definition~~Why are direct sums defined on sub-~~because the union is usually not a subspace so we use sums and keep it in subspacesand not sum of subsets?**spaces**