Vector Space and Linear Transformation

• Feb 2nd 2009, 07:25 AM
vincisonfire
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.
• Feb 2nd 2009, 11:58 AM
ThePerfectHacker
Quote:

Originally Posted by vincisonfire
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.