# 5.2

## Direct Sums

### Block matrices

- Talk about block matrices
- Start with the 1x1 block matrices
- Do bigger blocks
- Do block multiplication

### Direct sums equivalent definitions

- The following are equivalent:
- direct sum
- sums and unique way of writing 0
- unique way of writing any element
- union of ordered basis for summands is ordered basis
- exists order basis fof summands so that union is ordered basis.

- See book for proof
- Diagonalizabla if and only if there is a decomposition of V into eigenspaces.

# 5.3