- Cayley Hamilton is a thing

# 6.1 Inner Products

- Inner product is a generalization of a dot product, it only makes sense for subfields of C
- Complex conjugation is a thing
- The definition of inner product is that is is
- Linear in the first component
- (x,y) = conjugate of (y,x)
- Together these 2 conditions is called sesquilinear
- positive definite.

- Prop about inner product
- conjugate linear in 2nd component
- dotting with 0 gives zero.
- (x,x)=0 if and only if x=0
- (x,y)=(x,z) for all x implies y=z

- A vector space with an inner product is called an inner product space.
- Norms are a thing
- Prop about norms coming from inner products
- absolute values of scalars pull out
- norm zero iff zero
- Cauchy Schwarz
- Triangle

- Norm space
- Normalizing is a thing
- orthogonal is a thing
- Matrix representation.