• 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
  • Definition
• Normalizing is a thing
• orthogonal is a thing
• Matrix representation.