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

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    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)
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  2. #2
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    Quote Originally Posted by maximus101 View Post
    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
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