Is T a Linear Transformation from V to R1?

In summary, the conversation is about checking a proof for a linear transformation. The proof involves showing that T is a linear transformation from V into R1, where V is the space of all continuous functions on [0,1]. The proof involves showing that T satisfies the properties of linearity, namely T(f+g) = Tf + Tg and T(kf) = kTf. The conversation concludes with confirmation that the proof is correct and a clarification that V into R1 refers to the definite integral over a function in V giving a constant real number.
  • #1
discoverer02
138
1
I'd like to check my proof. It seems easy enough, but I'd like to make sure that I'm not missing anything:

If V is the space of all continuous functions on [0,1] and if
Tf = integral of f(x) from 0 to 1 for f in V, show that T is a linear transformation From V into R1.

Like I said the proof seems simple enough, but I just want to make sure I'm not missing anything that might be implied by "From V into R1."

T(f + g) = integral from 0 to 1[f(x) + g(x)]dx
= integral from 0 to 1 f(x)dx + integral from 0 to 1 g(x)dx
= Tf + Tg

T(kf) = integral from 0 to 1 kf(x)dx
= k*integral from 0 to1 f(x)dx
= kTf

there for T is a linear transformation.

I feel silly posting something this simple, but I'm just not absolutely sure that I'm not missing something.

Thanks as usual for all the help.
 
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  • #2
Yeah, that looks fine.

[tex]f:V \rightarrow\ R[/tex] just refers to the fact that the definite integral over a function in V will always give you a constant real number.
 
  • #3
Thanks Stevo.
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the original vector space. In simpler terms, it is a function that takes in a vector and outputs another vector, while maintaining the properties of linearity such as scaling and addition.

2. How are linear transformations represented?

Linear transformations can be represented in different ways, but the most common way is through a matrix. The columns of the matrix represent the output vectors for each basis vector in the input space. Another way to represent linear transformations is through a system of linear equations.

3. What is the difference between a linear transformation and a nonlinear transformation?

A linear transformation follows the properties of linearity, such as scaling and addition, while a nonlinear transformation does not. This means that a nonlinear transformation does not produce an output that is proportional to the input. Nonlinear transformations can be more complex and can include functions such as exponentials and logarithms.

4. How are linear transformations used in real life?

Linear transformations have a wide range of applications in fields such as physics, economics, and computer graphics. They can be used to model and solve problems involving linear systems, such as predicting the growth of a population or analyzing the flow of electricity in a circuit. In computer graphics, linear transformations are used to manipulate 2D and 3D objects, such as rotating or scaling them.

5. Can linear transformations change the dimension of a vector space?

Yes, linear transformations can change the dimension of a vector space. For example, a transformation that maps a 3D space to a 2D space will result in a dimension reduction. However, the transformation must still preserve the properties of linearity in order to be considered a linear transformation.

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