How this defines a linear transformation

• I
• Granger
In summary, in this conversation, the speaker is discussing a problem involving linear spaces and subspaces. They mention two linear transformations, S and T, that are defined on different subspaces but have the same output for values in their intersection. The speaker is asking for advice on how to define a linear transformation F that matches with both S and T. They suggest using F = S + T, but note that this may not work for objects in the intersection. The expert suggests considering direct sums and complements of subspaces in order to find a suitable definition for F.
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

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Granger said:
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

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 :)

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

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another, while preserving the structure of the original space. This means that the transformation preserves addition and scalar multiplication operations within the vector space.

2. How is a linear transformation defined?

A linear transformation can be defined as a function, denoted by T, that takes in a vector x and produces a transformed vector T(x). The transformation can be represented by a matrix, where each column represents the image of the corresponding basis vector in the original vector space.

3. What are the properties of a linear transformation?

A linear transformation must satisfy two properties: preservation of addition and preservation of scalar multiplication. This means that for any vectors u and v in the vector space and any scalar c, the transformation must follow the rules T(u+v) = T(u) + T(v) and T(cu) = cT(u).

4. How can linear transformations be visualized?

Linear transformations can be visualized through their effect on the vector space. For example, a linear transformation may rotate, reflect, or stretch the vector space in a certain direction. These transformations can be depicted using vector diagrams and matrices.

5. What are some real-life applications of linear transformations?

Linear transformations have many practical applications in fields such as physics, computer graphics, and economics. For example, in physics, linear transformations are used to represent the motion of objects. In computer graphics, they are used to manipulate and transform images. In economics, linear transformations are used to model economic systems and analyze data.

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