Math Help - Dimension and a describe basis of Hom_k(U,V)

1. Dimension and a describe basis of Hom_k(U,V)

If we let U and V be vector spaces of dimension n and m over K, and let $Hom_k$(U,V) be the vector space over K of all linear maps from U to V. What is the dimension and a basis of $Hom_k$(U,V)

2. Originally Posted by maximus101
If we let U and V be vector spaces of dimension n and m over K, and let $Hom_k$(U,V) be the vector space over K of all linear maps from U to V. What is the dimension and a basis of $Hom_k$(U,V)

Hint: choose basis $\{u_1,...,u_n\}\,,\,\{v_1,...,v_m\}$ of $U, V$ resp., and take a look at the linear

transformations determined for $1\leq i\leq n\,,\,1\leq j\leq m\,,\,T_{ij}:U\rightarrow V\,,\,\,T_{ij}u_i:=\delta_{ij}v_j$ ,

with $\delta_{ij}:=\left\{\begin{array}{ll}1&\mbox{ , if }i=j\\0&\mbox{ , if }i\neq k\end{array}\right.=$ the Kronecker delta .

Tonio