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Thread: 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 $\displaystyle
    \{\vec v_i : i \in I \} $ is a basis for V , listed without redundancy.
    Also suppose that for each $\displaystyle
    i \in I $, there exists $\displaystyle
    \{\vec w_i \in W \} $ (The list $\displaystyle
    \{\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 $\displaystyle T : V \rightarrow W$ such that $\displaystyle T\vec v_i = \vec w_i $ for every $\displaystyle i \in I $ .
    I was wondering if there was a less trivial way then to just say that there can be $\displaystyle T : V \rightarrow W$ such that $\displaystyle T\vec v_i = \vec w_i $ for every $\displaystyle 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 $\displaystyle
    \{\vec v_i : i \in I \} $ is a basis for V , listed without redundancy.
    Also suppose that for each $\displaystyle
    i \in I $, there exists $\displaystyle
    \{\vec w_i \in W \} $ (The list $\displaystyle
    \{\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 $\displaystyle T : V \rightarrow W$ such that $\displaystyle T\vec v_i = \vec w_i $ for every $\displaystyle i \in I $ .
    I was wondering if there was a less trivial way then to just say that there can be $\displaystyle T : V \rightarrow W$ such that $\displaystyle T\vec v_i = \vec w_i $ for every $\displaystyle 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 $\displaystyle T:V\to W$ in the following way. Let $\displaystyle v\in V$ then we can write $\displaystyle v = a_1 v_{(1)} + ... + a_n v_{(n)}$ where $\displaystyle v_{(j)} \in \{ v_i : i\in I\}$ . This is possible because $\displaystyle \{ v_i : i \in I\}$ is a basis. Furthermore, there is precisely one way of writing $\displaystyle v$ in terms of the basis elements. Now define $\displaystyle T(v) = \sum a_j w_{(j)}$. It follows that $\displaystyle T$ is a linear transformation with the desired properties.
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