Linear Algebra - Linear Transformation

[Eus]

Hi Ho! ^_^

I stuck when doing David C. Lay's Linear Algebra in Exercise 1.8 about Linear Transformation

I'm asked to determine whether these statements are correct.
Statement 1: A linear transformation is a special type of function.
Statement 2: The superposition principle is a physical description of a linear transformation.

My answer for statement 1:
The definition of linear transformation according to the book is as follows.
A transformation (or mapping) T is linear if:
1. T(c u + d v) = c T(u) + d T(v) for all u, v in the domain of T;
2. T(c u) = c T(u) for all u and all scalars c.

Therefore, according to me, statement 1 is correct because a linear transformation's function must satisfy the properties from the definition of linear transformations. Is this right?

My answer for statement 2:
According to the book, I rephrased it, the superposition principle is defined as the generalization of the definition of linear transformation
That is T(c1 v1 + ... + cp vp) = c1 T(v1) + ... + cp T(vp);
for v1...vp in the domain of T and c1...cp are scalars.

Therefore, in my opinion, statement 2 is true because a physical event can be determined to be linear if the "input" conditions can be expressed as a linear combination of such "input" and the system's response is the same linear combination of the responses to the individual "input". Is this correct?

Maybe you could provide me with a better answer for statement 1 or statement 2, please? ^^

Thank you very much!
Any help would be appreciated! ^_^

 

[HallsofIvy]

Therefore, according to me, statement 1 is correct because a linear transformation's function must satisfy the properties from the definition of linear transformations. Is this right?

A "linear transformation's function"? What is that? Aren't you assuming here that the linear transformation IS a function. Saying that some function "must satisfy the properties from the definition of linear transformation" is the wrong way around: you are showing that a certain function is a linear transformation.

You are correct that the statement is true but what you want to do to show that is state the definition of "function" and show that any linear transformation satisfies that definition.

Statement 2: The superposition principle is a physical description of a linear transformation.
Therefore, in my opinion, statement 2 is true because a physical event can be determined to be linear if the "input" conditions can be expressed as a linear combination of such "input" and the system's response is the same linear combination of the responses to the individual "input". Is this correct?

Once again, it seems to me you are going the "wrong way". Yes, a linear "physical event", by virtue of being linear, must satisfy the conditions for a linear transformation but the statement asserts that "The superposition principle is a physical description of a linear transformation." What, exactly, is a "physical description" (the question doesn't say anything about a "physical event") of a mathematical concept?

 

[thepatientmental]

Hi!

Yes, your answers for both statements 1 and 2 are correct. A linear transformation is indeed a special type of function that satisfies specific properties, and the superposition principle is a physical description of how a linear transformation behaves.

To provide a bit more context, linear transformations are important in linear algebra because they preserve the structure of vector spaces. This means that if you transform a vector in a vector space using a linear transformation, the resulting vector will still belong to that same vector space. This is why linear transformations are considered "special" and are often studied in depth in linear algebra.

The superposition principle, on the other hand, is a physical principle that states that the response of a system to a combination of inputs is the same as the sum of the responses to each individual input. This is essentially the same as the definition of a linear transformation, where the transformation of a linear combination of vectors is equal to the same linear combination of the transformed vectors.

I hope this helps clarify things a bit more. Keep up the good work with your studies! ^_^