Existence of an epimorphism

Question

Suppose $V$ and $W$ are finite-dimensional vector spaces over $F$ such that dim$_F(V)>$dim$_F(W)$. Is it true that

1. an epimorphism $f:V \rightarrow W$ exists?

2. a monomorphism $f:V \rightarrow W$ does not exist?

Answer 1

Choose basis $\{v_1,..,v_n\}$ to $V$ and $\{w_1,...,w_m\}$ to $W$ where $n>m$. Then for the epimorphism part you can just take, $$f(v_i)=w_i,\quad 1\leq i\leq m, \quad f(v_k)=0, m+1\leq k\leq n.$$ It easy to show that this is epimorphism.

On the other hand, for any $f:V\rightarrow W$ the imeges $$\{f(v_1),...,f(v_n)\}$$ are lineary dependent, and hence the kernal of $f$ is not trivial.