# Lin. Algebra - Sum of Dim. of Three Subspaces

• steelphantom
In summary, the conversation discusses a linear algebra question about the sum of subspaces in a finite-dimensional vector space. The main question is about proving the formula for the sum of dimension of three subspaces, with the additional option of providing a counterexample. The conversation concludes with the confirmation that the formula is indeed true and the suggestion to proceed with the proof.
steelphantom
Another linear algebra question! What a surprise!

## Homework Statement

If U1, U2, U3, are subspaces of a finite-dimensional vector space, then show

dim(U1 + U2 + U3) = dimU1 + dimU2 + dimU3 - dim(U1 $$\cap$$ U2) - dim(U1 $$\cap$$ U3) - dim(U2 $$\cap$$ U3) + dim(U1 $$\cap$$ U2 $$\cap$$ U3)

or give a counterexample.

## The Attempt at a Solution

I have the proof of the sum of the dimension of two subspaces in my book, so I would assume I would proceed in much the same way, but that "or give a counterexample" is making me just a little bit uneasy. I'm 90% sure that this is true, because basically the same formula holds for sets. Could anyone tell me if this is true before I proceed with my proof? It's going to be a long one if I use the same method the book did.

This has been asked more than once today, and I'll give you the same answer others have given. It is the same formula as for sets. And for the same reasons if you pick a compatible basis for the vector space. Proceed with your proof.

Dick said:
This has been asked more than once today, and I'll give you the same answer others have given. It is the same formula as for sets. And for the same reasons if you pick a compatible basis for the vector space. Proceed with your proof.

Thanks for the response. I finished my proof, but where was this question asked earlier today? I didn't see it in the Homework Help or Linear Algebra forums.

steelphantom said:
Thanks for the response. I finished my proof, but where was this question asked earlier today? I didn't see it in the Homework Help or Linear Algebra forums.

Hmmm. Now I can't find it. It may be under an obscure title. BTW, I didn't mean to say that you should have searched for other posts before asking. I was only saying my response wasn't original.

## 1. What is the definition of "Lin. Algebra - Sum of Dim. of Three Subspaces"?

The sum of dimensions of three subspaces in linear algebra refers to the total number of independent vectors required to span all three subspaces. It is calculated by adding the dimensions of each individual subspace and subtracting the dimension of their intersection.

## 2. Why is the sum of dimensions important in linear algebra?

The sum of dimensions allows us to determine the dimensionality of a vector space and understand the relationship between different subspaces. It also helps in solving systems of linear equations and finding the basis of a vector space.

## 3. How do you calculate the sum of dimensions of three subspaces?

To calculate the sum of dimensions, add the dimensions of each individual subspace and subtract the dimension of their intersection. For example, if subspace A has dimension 3, subspace B has dimension 2, and their intersection has dimension 1, then the sum of dimensions of these three subspaces is 3 + 2 - 1 = 4.

## 4. Are there any special cases where the sum of dimensions of three subspaces may be different?

Yes, there are two special cases. The first is when the intersection of the three subspaces is the zero vector, in which case the sum of dimensions will simply be the sum of the dimensions of the individual subspaces. The second case is when the intersection of the three subspaces is a subspace itself, in which case the sum of dimensions will be the sum of the dimensions of the individual subspaces minus the dimension of the intersection subspace.

## 5. How is the concept of sum of dimensions of three subspaces applied in real-world scenarios?

The concept of sum of dimensions is used in various areas of mathematics, engineering, and physics. For example, it is used in computer graphics to determine the dimensionality of a scene and in data analysis to understand the relationship between different variables. It is also used in linear control systems to analyze the stability of a system and in machine learning to reduce the dimensionality of data.

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