If we let U and V be vector spaces of dimension n and m over K, and let $\displaystyle Hom_k$(U,V) be the vector space over K of all linear maps from U to V. What is the dimension and a basis of $\displaystyle Hom_k$(U,V)
If we let U and V be vector spaces of dimension n and m over K, and let $\displaystyle Hom_k$(U,V) be the vector space over K of all linear maps from U to V. What is the dimension and a basis of $\displaystyle Hom_k$(U,V)
Hint: choose basis $\displaystyle \{u_1,...,u_n\}\,,\,\{v_1,...,v_m\} $ of $\displaystyle U, V$ resp., and take a look at the linear
transformations determined for $\displaystyle 1\leq i\leq n\,,\,1\leq j\leq m\,,\,T_{ij}:U\rightarrow V\,,\,\,T_{ij}u_i:=\delta_{ij}v_j$ ,
with $\displaystyle \delta_{ij}:=\left\{\begin{array}{ll}1&\mbox{ , if }i=j\\0&\mbox{ , if }i\neq k\end{array}\right.=$ the Kronecker delta .
Tonio