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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! ^_^

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# Homework Help: Linear Algebra - Linear Transformation

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