For a linear mapping F, how do I define F^2?

In summary, the conversation is about a linear mapping F where it is given that F is equal to F^2. The question is whether everything about F, such as dimension, kernel, and image, is the same for F^2 or if it just means that F applied twice to a vector v is equal to F applied once. The speaker mentions that this situation is similar to an "ordinary" projection in 3D.
  • #1
stgermaine
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Hi. This is a homework question, so I can't ask or give out too much info.

SO, there is a linear mapping F, and it is given that F=F^2.

Can I assume that everything about F, i.e. dimension, kernel, image, etc, is exactly the same for F^2? Or does it just mean that given a vector v, F(v) = F(F(v))?

Thank you
 
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  • #2
stgermaine said:
Hi. This is a homework question, so I can't ask or give out too much info.

SO, there is a linear mapping F, and it is given that F=F^2.

Can I assume that everything about F, i.e. dimension, kernel, image, etc, is exactly the same for F^2? Or does it just mean that given a vector v, F(v) = F(F(v))?

Thank you

It means F(v)=F(F(v)).
 
  • #3
hi stgermaine! :smile:

if F = F2, F is called a projection

think how an "ordinary" projection, say from 3D to a plane or a line, works :wink:
 
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Related to For a linear mapping F, how do I define F^2?

1. What is a linear mapping?

A linear mapping, also known as a linear transformation, is a mathematical function that maps a vector space to another vector space while preserving the structure of the original space. In simpler terms, it is a function that takes in a vector and outputs another vector in a linear manner.

2. What does F^2 mean in the context of a linear mapping?

In the context of a linear mapping, F^2 refers to the composition of the linear mapping F with itself. This means that the output of F is used as the input for another F, resulting in a new linear mapping.

3. How do I define F^2 for a linear mapping?

To define F^2, you can follow these steps:

  • Write out the definition of F in terms of matrix multiplication.
  • Replace each instance of F with the corresponding matrix representation.
  • Multiply the two matrices to get the matrix representation of F^2.

4. Can F^2 be defined for all linear mappings?

Yes, F^2 can be defined for all linear mappings. This is because the composition of two linear mappings will always result in another linear mapping.

5. What is the significance of defining F^2 for a linear mapping?

The significance of defining F^2 for a linear mapping is that it allows us to understand the behavior of the linear mapping over multiple iterations. This can be useful in various mathematical and practical applications, such as in understanding the stability of a system or in solving differential equations.

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