Prove U1+U2+U3 Theorem with Dimensions

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In summary, the conversation discusses how to prove that the dimensions of subsets U1, U2, and U3 of a finite set are related by the equation dim(U1+U2+U3) = dimU1 + dimU2 + dimU3 - dim(U1∩U2) - dim(U1∩U3) - dim(U2∩U3) + dim(U1∩U2∩U3). It is suggested to use the theorem dim(U1+U2) = dimU1 + dimU2 - dim(U1∩U2) to help prove this equation. The conversation then explores how to apply this theorem to V+U3, where V =
  • #1
pte419
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1. For subsets U1, U2, U3 of a finite set, prove that

dim(U1+U2+U3) = dimU1 + dimU2 + dimU3 - dim(U1∩U2) - dim(U1∩U3) - dim(U2∩U3) + dim(U1∩U2∩U3)



2. dim(U1+U2) = dimU1 + dimU2 - dim(U1∩U2)



3. I found that U1+U2 theorem in my book, and I think I should use that, but I'm not sure where to start...
 
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  • #2
Let V = U1 + U2. Now apply the theorem to V + U3.

Unless you are asked to prove 2 before proving 1. If this is the case please make it clear.
 
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  • #3
The two equations are also true, and easier to see, with the vectors spaces replaced by finite sets and the dimensions replaced by the sizes of the sets. It's possible, by picking certain bases, to make the two problems equivalent. But, as enumaelish hints, induction is probably easier.
 
  • #4
proof

I am only asked to prove equation one, but in doing that, I am allowed to use equation 2.
 
  • #5
Thanks guys, I've got one more question.

I did: dim(V+U3)
and I've ended up with:
dim(V+U3) = dimU1 + dimU2 - dim(U1∩U2) + dimU3 - dim(V∩U3)

Is there a property I can use to show that dim(V∩U3) is equivalent to the terms I still need to include for the proof? I can't find anything helpful in my book...
 
  • #6
[tex]dim(V\cap U_{3}) = dim((U_{1}+U_{2})\cap U_{3}) = dim((U_{1}\cap U_{3}) + (U_{2}\cap U_{3}))[/tex]

Break that term up again using the second part and you're done.
 

Related to Prove U1+U2+U3 Theorem with Dimensions

1. How do you prove the U1+U2+U3 Theorem?

The U1+U2+U3 Theorem states that the sum of three vectors in a vector space is equal to the sum of their individual components. To prove this, we start by defining our vector space and its dimensions. Then, we use the properties of vector addition and scalar multiplication to show that the sum of the vectors is equal to the sum of their components. This can be done using mathematical equations and logical reasoning.

2. What are the dimensions in the U1+U2+U3 Theorem?

The dimensions in the U1+U2+U3 Theorem refer to the number of components or parameters that make up each vector. For example, a vector in three-dimensional space would have three dimensions, while a vector in two-dimensional space would have two dimensions.

3. Can the U1+U2+U3 Theorem be applied to any vector space?

Yes, the U1+U2+U3 Theorem applies to any vector space, regardless of its dimensions or the type of vectors it contains. This theorem is a fundamental property of vector spaces and is applicable in many areas of mathematics and science.

4. How does the U1+U2+U3 Theorem relate to real-world applications?

The U1+U2+U3 Theorem has many real-world applications, especially in physics and engineering. For example, it can be used to calculate the net force on an object by adding the individual forces acting on it. It is also used in electrical circuits to find the total resistance of a circuit by adding the resistances of each component.

5. Is there a visual representation of the U1+U2+U3 Theorem?

Yes, the U1+U2+U3 Theorem can be visually represented using vector diagrams. These diagrams show the direction and magnitude of each vector and how they combine to form the resultant vector. This visual representation can help in understanding and applying the theorem in various contexts.

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