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Math Help - Vector Space and Linear Transformation

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    Senior Member vincisonfire's Avatar
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    Vector Space and Linear Transformation

    Let V and W be vector spaces over the field F ; suppose that <br />
\{\vec v_i : i \in I \} is a basis for V , listed without redundancy.
    Also suppose that for each <br />
 i \in I , there exists <br />
\{\vec w_i \in W \} (The list <br />
\{\vec w_i : i \in I \} might have redundancy; there’s no assumption on these, except that they are in W .) Show that there is a linear transformation  T : V \rightarrow W such that  T\vec v_i = \vec w_i for every  i \in I .
    I was wondering if there was a less trivial way then to just say that there can be  T : V \rightarrow W such that  T\vec v_i = \vec w_i for every  i \in I (which is reasonable in itself). And then to show that it respects linear transformation definition because W is a vector space.
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    Quote Originally Posted by vincisonfire View Post
    Let V and W be vector spaces over the field F ; suppose that <br />
\{\vec v_i : i \in I \} is a basis for V , listed without redundancy.
    Also suppose that for each <br />
 i \in I , there exists <br />
\{\vec w_i \in W \} (The list <br />
\{\vec w_i : i \in I \} might have redundancy; there’s no assumption on these, except that they are in W .) Show that there is a linear transformation  T : V \rightarrow W such that  T\vec v_i = \vec w_i for every  i \in I .
    I was wondering if there was a less trivial way then to just say that there can be  T : V \rightarrow W such that  T\vec v_i = \vec w_i for every  i \in I (which is reasonable in itself). And then to show that it respects linear transformation definition because W is a vector space.
    We will define T:V\to W in the following way. Let v\in V then we can write v = a_1 v_{(1)} + ... + a_n v_{(n)} where v_{(j)} \in \{ v_i : i\in I\} . This is possible because \{ v_i : i \in I\} is a basis. Furthermore, there is precisely one way of writing v in terms of the basis elements. Now define T(v) = \sum a_j w_{(j)}. It follows that T is a linear transformation with the desired properties.
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