Question

Show that if \(V\) is a finite dimensional vector space over \(F\) with subspaces \(W_{1}\) and \(W_{2},\) then $$\operatorname{dim}_{F}\left(W_{1}+W_{2}\right)=\operatorname{dim}_{F}\left(W_{1}\right)+\operatorname{dim}_{F}\left(W_{2}\right)-\operatorname{dim}_{F}\left(W_{1} \cap W_{2}\right)$$.

Short answer

Question: Show that the dimension of the sum of two subspaces is equal to the sum of their individual dimensions minus the dimension of their intersection. Short Answer: We have shown that the dimension of the sum of two subspaces, \(W_{1}+W_{2}\), is equal to the sum of their individual dimensions minus the dimension of their intersection, which is given by the theorem: $$\operatorname{dim}(W_{1}+W_{2})=\operatorname{dim}(W_{1})+\operatorname{dim}(W_{2})-\operatorname{dim}(W_{1}\cap W_{2})$$ This result was obtained by finding bases for the subspaces, their sum, and their intersection and then using the mentioned theorem to relate their dimensions.

Step-by-step solution

Define the subspaces W1 and W2

Let \(W_{1}\) and \(W_{2}\) be subspaces of a finite-dimensional vector space \(V\). That means they are both closed under vector addition and scalar multiplication, and each has a basis.

Find bases for W1 and W2

To determine dimensions, we first need bases for the subspaces. Let \(\{w_{1}, \ldots, w_{n}\}\) be a basis for \(W_{1}\) and \(\{v_{1}, \ldots, v_{m}\}\) be a basis for \(W_{2}\).

Construct the sum W1 + W2

The sum of two subspaces is defined as \(W_{1}+W_{2} = \{w + v \mid w \in W_{1}, v \in W_{2}\}\). This is again a subspace of \(V\). Since every element of \(W_{1} + W_{2}\) can be written as the sum of an element from each of the bases of \(W_{1}\) and \(W_{2}\), the set \(\{w_{1}, \ldots, w_{n}, v_{1}, \ldots, v_{m}\}\) spans \(W_{1}+W_{2}\).

Find a basis for the intersection W1 ∩ W2

The intersection \(W_{1} \cap W_{2}\) contains all vectors that are in both \(W_{1}\) and \(W_{2}\). It is again a subspace, and it has a basis. Let this basis be \(\{u_{1}, \ldots, u_{k}\}\).

Relate the dimensions using the theorem

There is a theorem that states for any two subspaces \(W_{1}\) and \(W_{2}\) of a vector space \(V\), the following holds: $$\operatorname{dim}(W_{1}+W_{2})=\operatorname{dim}(W_{1})+\operatorname{dim}(W_{2})-\operatorname{dim}(W_{1}\cap W_{2})$$ Since we have determined the bases for the sum and intersection of the subspaces, and we know the dimensions of these subspaces are equal to the number of elements in their respective bases, our task is complete. We have shown the theorem to be true for the given subspaces: $$\operatorname{dim}_{F}\left(W_{1}+W_{2}\right)=\operatorname{dim}_{F}\left(W_{1}\right)+\operatorname{dim}_{F}\left(W_{2}\right)-\operatorname{dim}_{F}\left(W_{1}\cap W_{2}\right)$$

Key concepts

Vector Space

In the realm of linear algebra, a vector space, also known as a linear space, is a fundamental concept that serves as the playground for vector operations. It's a collection of objects known as vectors which can be added together and multiplied by scalars, which are numbers from a given field, to produce another vector within the same space.

A vector space must satisfy several axioms, such as the existence of an additive identity (a zero vector, which when added to any vector in the space leaves the vector unchanged) and the distributive property (where multiplying a vector by the sum of scalars is the same as doing the operations separately and then adding the vectors). Two simple examples of vector spaces include the set of all 2-dimensional vectors (often visualized as arrows in a plane) and the set of all polynomials of a given degree. Every vector space is equipped with two operations: vector addition and scalar multiplication, which must adhere to specific rules to preserve structure.

Understanding what a vector space is helps in grasping the concepts of subspace, basis, and dimension, which define the structure and limitations of linear systems.

Basis of a Vector Space

The concept of a basis lies at the heart of understanding the composition and dimension of vector spaces. In essence, a basis of a vector space is a set of vectors that are linearly independent and span the entire space. This means that any vector within the space can be expressed as a unique combination (linear combination) of the basis vectors.

To relate this concept to familiar terms, think about constructing different shapes using a limited set of building blocks. The 'building blocks' in a vector space are the basis vectors, and they define the 'shapes' or vectors that can be built. In a 2-dimensional space, like a flat plane, a possible basis could consist of two non-parallel arrows, since they can be scaled and added to reach any point on the plane.

The number of vectors in the basis of a vector space is known as its dimension. For example, \( \mathbb{R}^3 \) has a dimension of three, typically represented by the basis vectors i, j, and k, which correspond to the three coordinate axes. If you're working in higher dimensions, the principle remains the same; you just have more 'directions' to consider.

Subspace Intersection

When learning about subspaces in the context of vector spaces, a particularly intriguing situation arises with their intersection. A subspace is essentially a smaller vector space tucked within a larger one, which itself adheres to all the same rules and axioms.

Now, consider two such subspaces, \(W_1\) and \(W_2\), both living within a larger vector space \(V\). The intersection of these subspaces, denoted \(W_1 \cap W_2\), contains all vectors that are members of both \(W_1\) and \(W_2\). This shared subspace is also itself a vector space, though potentially with a lower dimension. Understanding this is vital to working out the complex interplay of dimensions when you start combining subspaces.

In an educational exercise, if you're tasked with finding the dimension of this intersection, you're essentially counting the number of independent directions that are common to both \(W_1\) and \(W_2\). The dimensionality tells you how 'big' the intersection is. Just like in our earlier analogy with shapes and building blocks, the intersection represents a shared set of building blocks from both \(W_1\) and \(W_2\), and it's essential for understanding how vector spaces overlap and interact with each other.