Understanding the Relationship Between Dimensions

Oct 13, 2025 by windailys.com 72 views

Introduction

In linear algebra, the concept of dimension plays a crucial role in understanding the properties of vector spaces and their subspaces. Given a vector space V with a dimension of 17, and three subspaces U1, U2, and U3 with dimensions 8, 7, and 6 respectively, we are tasked with determining the maximal possible value of the dimension of the intersection of these subspaces. In this article, we will delve into the world of linear algebra and explore the relationship between the dimensions of subspaces and their intersections.

The Problem Statement

Let V be a vector space with a dimension of 17. We are given three subspaces U1, U2, and U3 with dimensions 8, 7, and 6 respectively. Additionally, we know that the dimension of the intersection of these subspaces is 4, i.e., dim(U1 ∩ U2 ∩ U3) = 4. Our goal is to find the maximal possible value of the dimension of the intersection of these subspaces.

Understanding the Relationship Between Dimensions

To approach this problem, we need to understand the relationship between the dimensions of subspaces and their intersections. The dimension of the intersection of two subspaces is given by the formula:

dim(U1 ∩ U2) = dim(U1) + dim(U2) - dim(U1 + U2)

where U1 + U2 is the sum of the subspaces U1 and U2. This formula can be extended to three subspaces as:

dim(U1 ∩ U2 ∩ U3) = dim(U1) + dim(U2) + dim(U3) - dim(U1 + U2) - dim(U1 + U3) - dim(U2 + U3) + dim(U1 + U2 + U3)

Applying the Formula

Using the given dimensions of the subspaces, we can plug in the values into the formula:

dim(U1 ∩ U2 ∩ U3) = 8 + 7 + 6 - dim(U1 + U2) - dim(U1 + U3) - dim(U2 + U3) + dim(U1 + U2 + U3)

We know that dim(U1 ∩ U2 ∩ U3) = 4, so we can set up the equation:

4 = 8 + 7 + 6 - dim(U1 + U2) - dim(U1 + U3) - dim(U2 + U3) + dim(U1 + U2 + U3)

Simplifying the Equation

Simplifying the equation, we get:

4 = 21 - dim(U1 + U2) - dim(U1 + U3) - dim(U2 + U3) + dim(U1 + U2 + U3)

The Maximal Possible Value of Dimension

To find the maximal possible value of the dimension of the intersection of the subspaces, we need to maximize the value of dim(U1 + U2 + U3). Since the dimension of the sum of two subspaces is less than or equal to the sum of their dimensions, we have:

dim(U1 + U2 + U3) ≤ dim(U1) + dim(U2) + dim(U3)

= 8 + 7 + 6

= 21

Therefore, the maximal possible value of the dimension of the intersection of the subspaces is:

dim(U1 ∩ U2 ∩ U3) = 4

Conclusion

In conclusion, we have shown that the maximal possible value of the dimension of the intersection of three subspaces U1, U2, and U3 with dimensions 8, 7, and 6 respectively, and a dimension of the intersection of 4, is 4. This result is a direct consequence of the relationship between the dimensions of subspaces and their intersections, and the properties of vector spaces.

References

[1] Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice Hall.
[2] Lang, S. (1987). Linear Algebra. Addison-Wesley.
[3] Axler, S. (1997). Linear Algebra Done Right. Springer-Verlag.

Introduction

In our previous article, we explored the relationship between the dimensions of subspaces and their intersections, and determined the maximal possible value of the dimension of the intersection of three subspaces U1, U2, and U3 with dimensions 8, 7, and 6 respectively, and a dimension of the intersection of 4. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the relationship between the dimensions of subspaces and their intersections?

A: The dimension of the intersection of two subspaces is given by the formula:

dim(U1 ∩ U2) = dim(U1) + dim(U2) - dim(U1 + U2)

where U1 + U2 is the sum of the subspaces U1 and U2. This formula can be extended to three subspaces as:

dim(U1 ∩ U2 ∩ U3) = dim(U1) + dim(U2) + dim(U3) - dim(U1 + U2) - dim(U1 + U3) - dim(U2 + U3) + dim(U1 + U2 + U3)

Q: How do I determine the dimension of the intersection of two subspaces?

A: To determine the dimension of the intersection of two subspaces, you need to know the dimensions of the subspaces and the dimension of their sum. You can use the formula:

dim(U1 ∩ U2) = dim(U1) + dim(U2) - dim(U1 + U2)

Q: What is the maximal possible value of the dimension of the intersection of three subspaces?

A: The maximal possible value of the dimension of the intersection of three subspaces U1, U2, and U3 with dimensions 8, 7, and 6 respectively, and a dimension of the intersection of 4, is 4.

Q: How do I determine the dimension of the sum of two subspaces?

A: To determine the dimension of the sum of two subspaces, you need to know the dimensions of the subspaces. The dimension of the sum of two subspaces is less than or equal to the sum of their dimensions, i.e.,

dim(U1 + U2) ≤ dim(U1) + dim(U2)

Q: What is the relationship between the dimension of a subspace and the dimension of its complement?

A: The dimension of a subspace and the dimension of its complement are related by the formula:

dim(U) + dim(U^c) = dim(V)

where U is a subspace of V, and U^c is the complement of U.

Q: How do I determine the dimension of the complement of a subspace?

A: To determine the dimension of the complement of a subspace, you need to know the dimension of the subspace and the dimension of the vector space. You can use the formula:

dim(U^c) = dim(V) - dim(U)

Conclusion

In conclusion, we have answered some frequently asked questions related to the maximal possible value of the dimension of the intersection of three subspaces. We hope that this article has been helpful in clarifying the concepts and formulas related to this topic.

References

[1] Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice Hall.
[2] Lang, S. (1987). Linear Algebra. Addison-Wesley.
[3] Axler, S. (1997). Linear Algebra Done Right. Springer-Verlag.