by **HallsofIvy** » Thu Mar 14, 2019 9:01 pm

Start from the definitions! A subset, E, of vector space, V, is a subspace if and only if it is "closed under vector addition" and "closed under scalar multiplication". That is, if u and v are both in E then u+ v is also in E and if v is in E and a is a scalar then av is also in E. Also, in linear algebra, "endomorphisms" are linear functions.

Here, E is the set of all v such that [tex]f(v)= \lambda v[/tex].

If u and v are in E, so that [tex]f(u)= \lambda u[/tex] and [tex]f(v)= \lambda v[/tex], then [tex]f(u+ v)= f(u)+ f(v)= \lambda u+ \lambda v= \lambda(u+ v)[/tex]. Therefore u+ v is in E.

If v is in E and a is a scalar then [tex]f(av)= af(v)= a \lambda v= \lambda (av)[/tex]. Therefore av is in E.