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 subspace sum of subsets direct sum Results and Their Proofs checking for subspaces simplified check for subspace sum of subspaces is the smallest subspace with both subspaces creating direct sums a sum of subsets is a direct sum IFF there is only one way to write 0 a sum of subsets is only a direct sum IFF their intersection is the set containing 0 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-spaces and not sum of subsets? because the union is usually not a subspace so we use sums and keep it in subspaces