Linear Algebra - Linear Transformation

  • Thread starter Eus
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Eus
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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! ^_^
 

Answers and Replies

  • #2
HallsofIvy
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Eus said:
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?
 

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