Tricky subspace & intersection Problem

In summary, the conversation discusses the problem of proving that W_1 ∩ (W_2 + ( W_1 ∩ W_3)) = (W_1 ∩ W_2) + (W_1 ∩ W_3) for subspaces W_1, W_2, W_3 of a vector space V. The conversation then moves on to discussing Dedekind's law and how it may apply to the problem. Finally, the conversation concludes with a solution being reached and a question about the validity of the equality and any potential restrictions.
  • #1
aeronautical
34
0

Homework Statement



I am trying to solve this problem:
Let W_1, W_2, W_3 be subspaces of a vector space, V.
Prove that W_1 ∩ (W_2 + ( W_1 ∩ W_3)) = (W_1 ∩ W_2) + (W_1 ∩ W_3).
Can someone help me show this? I have tried using Dedekind's law, but not sure it that is the way to go.


The attempt at a solution

I tried with in my mind very trivial case...can somebody please show me a more detailed solution with more steps?

This is what I did...Since a subspace is a set, the laws of set operations apply. I assume (not sure if this is a valid assumption) that + here is the same as "union".

Now intersection is distributive over union,
i.e. a∩(b+c) = a∩b + a∩c
so in this case,
W_1 ∩ (W_2 + ( W_1 ∩ W_3))
= (W_1 ∩ W_2) + (W_1 ∩ (W_1 ∩ W_3))
In the second term, I use the properties that intersection is associative, and W_1 ∩ W_1 = W_1, and that term becomes W_1 ∩ W_3 which proves the required result.


Now can anyone answer if this always holds and please show me a more detailed solution with more steps that would make more sense? Thanks...
 
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  • #2
I haven't thought about how to do the question, but
aeronautical said:
I assume (not sure if this is a valid assumption) that + here is the same as "union".

is not a valid assumption. Have you looked in your notes or text for a definition?
 
  • #3
George Jones said:
I haven't thought about how to do the question, but


is not a valid assumption. Have you looked in your notes or text for a definition?

Yes I did...but unfortunately, I am totally lost and confused now...
 
  • #4
aeronautical said:
Yes I did...but unfortunately, I am totally lost and confused now...

Write down Dedekind's law (in a post).
 
  • #5
George Jones said:
Write down Dedekind's law (in a post).

V ∩ (U + W)= U + (V ∩ W) this is what I got...
 
  • #6
aeronautical said:
V ∩ (U + W)= U + (V ∩ W) this is what I got...

Is there a condition that U and V are required to satisfy?
 
  • #7
U, V and W, are just subsets of a set S, right?
 
  • #9
In my question those are all the conditions specified. Do you mean that U <= V? How does, this help me understand whether the LHS is equal to the RHS? The real question is thus, when is this valid, and how I can get the RHS from LHS?
 
  • #10
The LHS of your question and the LHS of Dedekind's law look similar.

[edit}

To make them the same, take [itex]V = W_1[/itex], and either [itex]U = W_2[/itex] and [itex]W = W_1 \cap W_3[/itex], or [itex]W = W_2[/itex] and [itex]U = W_1 \cap W_3[/itex].

[/edit]

With respect to Dedekind's law, does either of these choices allow you to show the desired result?
 
  • #11
I don't think so:
The first one results in:
RHS = W_1 + (W_2 + W_1 ∩ W_3)

The second one results in:
W_1 + (W_1 ∩ W_3 ∩ W_2)

What should I do now?? What conclusion can I take away from this?
 
  • #12
Take a look at the second choice more carefully.
 
  • #13
I realize that for the equality to hold then:

W_1 + (W_1 ∩ W_3 ∩ W_2) = W_1 ∩ (W_2 + ( W_1 ∩ W_3))

However I can not see how... Could you please guide?
 
  • #14
I'm a little confused.

With the second of my choices in post #10 (note the edit):

what are V, U , and W;

what is the LHS of Dedekind's laws;

what is the RHS of Dedekind's laws?
 
  • #15
I just noticed the edit:
Using the edited U, V, W from post 10 I get that:
W_1 ∩ (W_1 ∩ W_3 +W_2) = (W_1 ∩ W_3) + (W_1 ∩ W_2)

Hence, I have shown the equality in the original problem statement (Thank you).

One question remains and that is... Is this equality ALWAYS valid? Are there any restrictions? I thought the condition in the original Dedekind's law with U <= V could be one. Are there others?
 
  • #16
Dear George,
Could you please help me out regarding my previous questions (In Post #15)? Thank you...
 

1. What is a tricky subspace & intersection problem?

A tricky subspace & intersection problem is a mathematical problem that involves finding the intersection between two or more subspaces, which are subsets of a larger vector space. These types of problems can be challenging because they require a deep understanding of vector spaces, linear algebra, and geometry.

2. What are some common techniques for solving tricky subspace & intersection problems?

There are several techniques that can be used to solve tricky subspace & intersection problems, including:

  • Geometric approach: This involves visualizing the subspaces and their intersection in 2D or 3D space to gain a better understanding of the problem.
  • Algebraic approach: This involves using algebraic equations and operations to solve the problem, such as finding the null space or using the rank-nullity theorem.
  • Orthogonal projection: This technique involves projecting one subspace onto another to find the intersection.
  • Gram-Schmidt process: This is a method for finding an orthonormal basis for a subspace, which can be useful in solving tricky intersection problems.

3. Why are tricky subspace & intersection problems important in science and mathematics?

Tricky subspace & intersection problems are important in science and mathematics because they have applications in various fields, including physics, engineering, and computer science. These problems help to develop critical thinking and problem-solving skills, and they also provide a deeper understanding of vector spaces and their properties.

4. Can tricky subspace & intersection problems have real-world applications?

Yes, tricky subspace & intersection problems can have real-world applications. For example, in physics, these types of problems can be used to analyze the motion of objects in space or to determine the forces acting on an object. In engineering, these problems can be used to design efficient and stable structures. In computer science, they can be used in image and signal processing algorithms.

5. How can one improve their skills in solving tricky subspace & intersection problems?

Improving skills in solving tricky subspace & intersection problems can be achieved through practice and understanding the underlying mathematical concepts. It is also helpful to explore different techniques and approaches for solving these problems and to seek guidance from experts or study materials. Additionally, developing a strong foundation in linear algebra and vector spaces can greatly improve problem-solving skills in this area.

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