Dimension proof of the intersection of 3 subspaces

• harvesl
In summary, the conversation discusses the question of whether the dimension of the intersection of three distinct subspaces of dimension n-1 in a vector space of dimension n is always equal to n-3. The person asking the question provides their thoughts on the matter but is unsure of their solution. They also mention a previous question that they were able to prove. The expert confirms that the person is on the right track and provides reassurance.
harvesl

Homework Statement

Assume $V = \mathbb{R}^n$ where $n \geq 3$. Suppose that $U,W,X$ are three distinct subspaces of dimension $n-1$; is it true then that $dim(U \cap W \cap X) = n-3$? Either give a proof, or find a counterexample.

The Attempt at a Solution

The question previous to this was showing that for subspaces $U,W$ of dimension $n-1 \longrightarrow dim(U \cap W) = n - 2$ which I was able to prove fine. Now, I'm thinking this is true, because $W \cap X$ will be a subspace of dimension $n-2$ so if we set $W \cap X = Z$, where $Z$ is a subspace we turn this problem into $dim(U \cap Z)$ and since all three were distinct we should have $dim(U \cap Z) = n-3$ but I'm not entirely sure.

Thanks!

You seem to be on the right track.

voko said:
You seem to be on the right track.

Thanks.

1. What is the dimension proof of the intersection of 3 subspaces?

The dimension proof of the intersection of 3 subspaces is a mathematical concept that shows the relationship between the dimensions of three subspaces and their intersection. It is used to determine the maximum possible dimension of the intersection of the subspaces.

2. How is the dimension proof of the intersection of 3 subspaces calculated?

The dimension proof is calculated by finding the dimensions of each individual subspace, and then using the formula dim(V1 ∩ V2 ∩ V3) = dim(V1) + dim(V2) + dim(V3) - dim(V1 ∪ V2) - dim(V1 ∪ V3) - dim(V2 ∪ V3) + dim(V1 ∪ V2 ∪ V3). This formula takes into account any overlaps between the subspaces.

3. Why is the dimension proof of the intersection of 3 subspaces important?

This proof is important because it allows us to determine the maximum possible dimension of the intersection of three subspaces, which can provide valuable insights in fields such as linear algebra, computer science, and physics.

4. Can this proof be applied to more than 3 subspaces?

Yes, this proof can be generalized to any number of subspaces. The formula for calculating the dimension of the intersection would involve adding or subtracting the dimensions of the subspaces depending on their overlaps.

5. Are there any real-world applications of the dimension proof of the intersection of 3 subspaces?

Yes, this proof has practical applications in fields such as signal processing, image processing, and data compression. It is also commonly used in linear regression analysis and machine learning algorithms.

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