6Endomorphisms

IB Linear Algebra

6 Endomorphisms

Endomorphisms are linear maps from a vector space

V

to itself. One might

wonder — why would we want to study these linear maps in particular, when

we can just work with arbitrary linear maps from any space to any other space?

When we work with arbitrary linear maps, we are free to choose any basis

for the domain, and any basis for the co-domain, since it doesn’t make sense to

require they have the “same” basis. Then we proved that by choosing the right

bases, we can put matrices into a nice form with only 1’s in the diagonal.

However, when working with endomorphisms, we can require ourselves to use

the same basis for the domain and co-domain, and there is much more we can

say. One major objective is to classify all matrices up to similarity, where two

matrices are similar if they represent the same endomorphism under different

bases.