Question

From the previous exercise, one might be tempted to think that a more general "inclusion/exclusion principle" for dimension holds. Determine if the following statement is true or false: if \(V\) is a finite dimensional vector space over \(F\) with subspaces \(W_{1}, W_{2},\) and \(W_{3},\) then $$\begin{aligned} \operatorname{dim}_{F}\left(W_{1}\right.&\left.+W_{2}+W_{3}\right)=\operatorname{dim}_{F}\left(W_{1}\right)+\operatorname{dim}_{F}\left(W_{2}\right)+\operatorname{dim}_{F}\left(W_{3}\right) \\\ &-\operatorname{dim}_{F}\left(W_{1} \cap W_{2}\right)-\operatorname{dim}_{F}\left(W_{1} \cap W_{3}\right)-\operatorname{dim}_{F}\left(W_{2} \cap W_{3}\right) \\ &+\operatorname{dim}_{F}\left(W_{1} \cap W_{2} \cap W_{3}\right) \end{aligned}.$$

Short answer

Answer: Yes, based on the example we explored with \(\mathbb{R}^3\) and its three subspaces, the given equation holds true for the inclusion-exclusion principle for dimensions of these subspaces.

Step-by-step solution

Define Counterexample Subspaces

Let's take \(V\) to be a 3-dimensional vector space \(\mathbb{R}^3\) with \(F=\mathbb{R}\) as the field of scalars. We can construct three subspaces for this vector space as follows: - \(W_1 =\) Span{\((1, 0, 0)\)} - \(W_2 =\) Span{\((0, 1, 0)\)} - \(W_3 =\) Span{\((0, 0, 1)\)} Now, let's calculate the dimensions of the individual subspaces and their intersections.

Calculate Individual Dimensions

The dimensions of the given subspaces are: - \(\operatorname{dim}_{\mathbb{R}}(W_1) = 1\) because \(W_1\) has 1 basis vector. - \(\operatorname{dim}_{\mathbb{R}}(W_2) = 1\) because \(W_2\) has 1 basis vector. - \(\operatorname{dim}_{\mathbb{R}}(W_3) = 1\) because \(W_3\) has 1 basis vector. Now let's compute the dimensions of the intersections of these subspaces.

Calculate Intersection Dimensions

The intersections of the subspaces are as follows: - \(\operatorname{dim}_{\mathbb{R}}(W_1 \cap W_2) = 0\) because \(W_1\) and \(W_2\) have no common vectors (except the zero vector). - \(\operatorname{dim}_{\mathbb{R}}(W_1 \cap W_3) = 0\) because \(W_1\) and \(W_3\) have no common vectors (except the zero vector). - \(\operatorname{dim}_{\mathbb{R}}(W_2 \cap W_3) = 0\) because \(W_2\) and \(W_3\) have no common vectors (except the zero vector). - \(\operatorname{dim}_{\mathbb{R}}(W_1 \cap W_2 \cap W_3) = 0\) because the three subspaces have no common vectors (except the zero vector). Now we can evaluate the given equation using the calculated dimensions.

Evaluate the Given Equation

According to the given equation, we have: $$\begin{aligned} \operatorname{dim}_{\mathbb{R}}(W_1 + W_2 + W_3) = &\operatorname{dim}_{\mathbb{R}}(W_1) + \operatorname{dim}_{\mathbb{R}}(W_2) + \operatorname{dim}_{\mathbb{R}}(W_3) \\ &- \operatorname{dim}_{\mathbb{R}}(W_1 \cap W_2) - \operatorname{dim}_{\mathbb{R}}(W_1 \cap W_3) - \operatorname{dim}_{\mathbb{R}}(W_2 \cap W_3) \\ &+ \operatorname{dim}_{\mathbb{R}}(W_1 \cap W_2 \cap W_3) \end{aligned}$$ Plugging in the dimensions we calculated in steps 2 and 3, we get: $$\begin{aligned} \operatorname{dim}_{\mathbb{R}}(W_1 + W_2 + W_3) = &1 + 1 + 1 \\ &- 0 - 0 - 0 \\ &+ 0 \end{aligned}$$ This simplifies to: $$\operatorname{dim}_{\mathbb{R}}(W_1 + W_2 + W_3) = 3$$ However, we know that \(\operatorname{dim}_{\mathbb{R}}(W_1 + W_2 + W_3) = \operatorname{dim}_{\mathbb{R}}(\mathbb{R}^3) = 3\). As our calculated dimensions match the dimensions of the given vector space, we can conclude that the given statement is indeed true. Thus, the inclusion-exclusion principle for dimension holds for this example.

Key concepts

Finite Dimensional Vector Space

A finite dimensional vector space is a vector space that has a finite basis. The basis of a vector space is a set of vectors that are linearly independent and span the entire space. This concept is pivotal because it allows us to define the dimension of a vector space, essentially denoting the number of vectors in any basis for the space.

In simpler terms, a finite dimensional vector space could be thought of as a space where you can count the directions available for travel without ever running out of numbers. For example, in a 3-dimensional space, often represented by \(\mathbb{R}^3\), there are three basic directions, X, Y, and Z, corresponding to forward/backward, left/right, and up/down. Any point in this space can be reached by moving the right amount in each of these three directions.

The importance of vector spaces in mathematics and physics is immense, as they offer a way to abstractly model different phenomena, allowing for operations such as addition of vectors and multiplication by scalars. In fields like engineering, economics, and natural sciences, finite dimensional vector spaces are foundational because their structure is easy to understand and predict due to their limited number of dimensions.

Subspace Intersections

The concept of subspace intersections comes into play when looking at commonalities between subspaces within a vector space. Subspaces are themselves vector spaces that satisfy two criteria: They must contain the zero vector and be closed under vector addition and scalar multiplication.

When we look at the intersection of subspaces, we're essentially looking for a new subspace made up of vectors that every original subspace has in common. Consider a classroom, two clubs might share it for their meetings, the intersection would be the time slots when the room is booked by both clubs simultaneously. In vector terms, for our vector space \(\mathbb{R}^3\), if we have three subspaces corresponding to the directions X, Y, and Z, their intersections are conceptually where these directions would meet.

When the subspaces intersect only at the zero vector, they are said to be in direct sum; that is, each vector in the sum of the subspaces can be uniquely written as the sum of vectors from each subspace. However, if they share more than the zero vector, this intersection must be considered to avoid overcounting dimensions when we talk about the combined dimensions of these subspaces.

Vector Space Dimension

The vector space dimension is a fundamental property that measures the size of the vector space in terms of how many base vectors are needed to span the space. This is analogous to figuring out the minimum number of unique building blocks needed to construct any element within the space.

In the context of the exercise, the dimension of a vector space can be seen as the number of lines in a recipe that defines how to cook any dish from a cooking book, representing all possible dishes (vectors) you can prepare. Each line corresponds to a key ingredient or step without which the recipe would be incomplete, similar to how each basis vector is essential for the completeness of the space.

Understanding dimensions of subspaces and their intersections is critical because when we combine (add) subspaces, we might accidentally count dimensions multiple times if they share vectors. Thus, the dimension of the sum is corrected by subtracting the dimensions of pairwise intersections and adding back the dimension of the triple intersection if all three share vectors, following an analogous approach to the inclusion-exclusion principle in set theory.