Linear Algebra - Linear Transformation

How is the superposition principle "physical"? I think you need to think more about that and about what "superposition principle" actually means.
  • #1
Eus
94
0
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! ^_^
 
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  • #2
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?
 
  • #3


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! ^_^
 

Related to Linear Algebra - Linear Transformation

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves the operations of vector addition and scalar multiplication. In other words, the output of a linear transformation is a linear combination of the input vectors.

2. How is linear algebra used in linear transformation?

Linear algebra is used to represent and manipulate linear transformations in a mathematical and systematic way. It provides tools such as matrices, vectors, and linear transformations to solve problems and analyze data in a wide range of fields including physics, engineering, and computer science.

3. What are some applications of linear transformation?

Linear transformations have many practical applications, including in computer graphics, image processing, data compression, and machine learning. They are also used in physics to describe the motion of objects and in economics to model relationships between variables.

4. How can I determine if a transformation is linear?

A transformation is linear if it satisfies two properties: preservation of vector addition and preservation of scalar multiplication. In simpler terms, this means that the output of the transformation must be the same when the operations are performed on the input vectors individually or together.

5. What is the relationship between linear transformation and matrices?

Matrices are a way of representing linear transformations in a compact and efficient manner. Each column of a matrix represents the image of a standard basis vector under the transformation. By performing operations on matrices, such as multiplication, we can easily compute the output of a linear transformation for any input vector.

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