# Prove infinitely many left inverses

## Homework Statement

Let V be a vector space over K. Let L(V) be the set of all linear maps V->V. Prove that L(V) is a ring under the operations:
f+g:x -> f(x)+g(x) and fg:x -> f(g(x))

Now, let V=U+W be the direct sum of two vector spaces over K such that the dimension of both U and W are countable. Then V has countable dimension. Choosing a linear bijection between U and V gives us an element f:V->U of L(V). Prove that there are infinitely many $$x \in R = L(V)$$ such that xf=1_R. Prove that there is no $$y \in R$$ such that fy=1_R.

## Homework Equations

Direct sum of two vector spaces U and W is the set U+W of pairs of vectors (u,w) in U and W with operations:
a(u,w)+b(u'+w')=(au+bu',aw+bw')

## The Attempt at a Solution

For the first bit, I managed to show that L(V) is indeed a ring. In the second part, I'm not sure how to approach this problem. Should I define a bijective function f such that xf=1_R? Also, is linear bijection essentially means an isomorphism?