------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.