Axler 1.C

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