MHB How to define this linear transformation

[Granger]

> Admit that $V$ is a linear space about $\mathbb{R}$ and that $U$ and $W$ are subspaces of $V$. Suppose that $S: U \rightarrow Y$ and $T: W \rightarrow Y$ are two linear transformations that satisfy the property:

> $(\forall x \in U \cap W)$ $S(x)=T(x)$

> Define a linear transformation $F$: $ U+W \rightarrow Y$ that matches with S for values in U and matches with T with values in W.

My thought is to choose the linear transformation $F=S+T$ because it will be the union of both transformation, right? But now I know this is incorrect...

Can someone give an hint on how to approach the problem (without using matrices...)?

I know that the function might be equal to S for the vectors that belong to U and equal to T for vector that belong to V... But how do I get there and write a linear transformation?

 

[Opalg]

> Admit that $V$ is a linear space about $\mathbb{R}$ and that $U$ and $W$ are subspaces of $V$. Suppose that $S: U \rightarrow Y$ and $T: W \rightarrow Y$ are two linear transformations that satisfy the property:

> $(\forall x \in U \cap W)$ $S(x)=T(x)$

> Define a linear transformation $F$: $ U+W \rightarrow Y$ that matches with S for values in U and matches with T with values in W.

My thought is to choose the linear transformation $F=S+T$ because it will be the union of both transformation, right? But now I know this is incorrect...

Can someone give an hint on how to approach the problem (without using matrices...)?

I know that the function might be equal to S for the vectors that belong to U and equal to T for vector that belong to V... But how do I get there and write a linear transformation?

A vector $x$ in $U+W$ has to be of the form $x = u+w$, where $u\in U$ and $w\in W$. The natural way to define $F$ is $F(x) = S(u) + T(w)$. The thing you have to be careful about is to show that this $F$ is well-defined. In other words, suppose that $x$ can be expressed as an element of $U+W$ in more than one way, say $x = u_1 + w_1$ and also $x = u_2 + w_2$. In order for the definition of $F$ to make sense, it should be true that both expressions for $x$ give rise to the same value for $F(x)$. So you have to show that $S(u_1) + T(w_1) = S(u_2) + T(w_2).$ Can you do that?