How this defines a 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 we have the problem that when our objects are in the intersection of both space we will get to F = 2S instead of S

 

[Stephen Tashi]

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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.

Are you asking whether it is possible to define such a linear transformaton ?

My thought is to choose the linear transformation $F=S+T$ because it will be the union of both transformation, right? But we have the problem that when our objects are in the intersection of both space we will get to F = 2S instead of S

 

[Granger]

Hello Stephen!

Thanks, I edited my question.

No, I know it's possible to define the linear transformation I'm asking what should be the linear transformation and how can I get there :)

 

[jbunniii]

Do you know about direct sums, and complements of subspaces?