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Chapter 2: Problem 7

Sei \(F: V \rightarrow W\) linear und \(U \subset W\) ein Untervektorraum. Zeigen Sie, dass dann $$ \operatorname{dim} F^{-1}(U)=\operatorname{dim}(U \cap \operatorname{Im} F)+\operatorname{dimKer} F $$

Short Answer

Expert verified
\(\operatorname{dim} F^{-1}(U) = \operatorname{dim}(U \cap \operatorname{Im} F) + \operatorname{dimKer} F\) is shown by considering the decomposition of elements in \(F^{-1}(U)\).

Step by step solution

01

Understand the Concepts

First, recall that if \(F: V \rightarrow W\) is a linear map, then the image \(\operatorname{Im} F\) of \(F\) is the set of all vectors \(w \in W\) such that \(w = F(v)\) for some \(v \in V\). The kernel \(\operatorname{Ker} F\) is the set of all vectors \(v \in V\) such that \(F(v) = 0\). An inverse image \(F^{-1}(U)\) is the set of all vectors \(v \in V\) such that \(F(v) \in U\). \(U \cap \operatorname{Im} F\) is the intersection of \(U\) with the image of \(F\).
02

Use the Rank-Nullity Theorem

The rank-nullity theorem states that if \(F: V \rightarrow W\) is a linear map, then \[ \operatorname{dim} V = \operatorname{dim} \operatorname{Im} F + \operatorname{dim} \operatorname{Ker} F. \] This means that the dimension of the vector space \(V\) is the sum of the dimensions of the image and the kernel of \(F\).
03

Consider Pre-image and Intersection

The set \(F^{-1}(U)\) consists of all vectors in \(V\) that are mapped into \(U\). Since \(U\) is a subspace of \(W\), the intersection \(U \cap \operatorname{Im} F\) is a subspace of \(\operatorname{Im} F\), and the dimension \(\operatorname{dim}(U \cap \operatorname{Im} F)\) can be computed.
04

Relate the Dimensions

Any vector \(v \in F^{-1}(U)\) can be written as \(v = k + x\), where \(k \in \operatorname{Ker} F\) and \(x\) is a vector such that \(F(x) \in U \cap \operatorname{Im} F\). Therefore, each element of \(F^{-1}(U)\) corresponds to a unique element of \(\operatorname{Ker} F\) and \(U \cap \operatorname{Im} F\).
05

Conclusion of Dimension Equality

We have established that every element of \(F^{-1}(U)\) can be uniquely represented as a sum of an element from \(\operatorname{Ker} F\) and an element from \(U \cap \operatorname{Im} F\). Thus, the dimension formula \(\operatorname{dim} F^{-1}(U) = \operatorname{dim}(U \cap \operatorname{Im} F) + \operatorname{dimKer} F\) holds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rank-Nullity Theorem
The Rank-Nullity Theorem is a fundamental principle in linear algebra. It provides a clear relationship between important subspaces of a linear map. Imagine you have a linear map \(F: V \rightarrow W\), where \(V\) and \(W\) are vector spaces. The theorem states:

\[ \operatorname{dim} V = \operatorname{dim} \operatorname{Im} F + \operatorname{dim} \operatorname{Ker} F. \]
  • Dimension of \(V\): Total number of basis vectors in space \(V\).
  • Image \(\operatorname{Im} F\): Consists of all vectors in \(W\) that can be expressed as \(F(v)\) for some \(v\) in \(V\).
  • Kernel \(\operatorname{Ker} F\): Consists of all vectors \(v\) in \(V\) such that their image under \(F\) is the zero vector.
In simple terms, the sum of the dimensions of the image and the kernel is equal to the dimension of the original space \(V\). This theorem helps us understand the distribution of vectors across the kernel and image of \(F\). It is incredibly useful when analyzing the behavior of functions and transformations.
Kernel and Image
A good way to begin understanding the kernel and image is by considering the roles they play in a linear map. Think of these as two distinct parts that work together in mapping vectors.

  • Kernel (Ker F): The kernel \( \operatorname{Ker} F \) is the set of vectors in \( V \) that are mapped to zero in \( W \) by the function \( F \). In mathematical language, these are solutions to the equation \( F(v) = 0 \).
  • Image (Im F): The image \( \operatorname{Im} F \) of \( F\) is the set of vectors in \( W \) that can be obtained as \( F(v) \) for some \( v \) in \( V \). It represents all possible outputs of the linear map.
These concepts are crucial when analyzing linear transformations, particularly when we are interested in understanding what vectors get "nullified" or what range of outputs we can achieve from the transformation. They allow us to envision how the entire space is transformed.
Subspace Intersection
When talking about subspace intersection, we're dealing with the concept of overlapping areas within subspaces created by a linear map. Let's simplify what this means.

  • Intersection \(U \cap \operatorname{Im} F\): This represents the set of vectors that are part of both subspace \(U\) and the image of \(F\).
  • Properties of Intersection: By nature, the intersection of two subspaces is itself a subspace. This allows us to explore shared dimensions and components between multiple vector spaces.
In the context of the exercise, we're interested in how a particular subspace \(U\) overlaps with the results produced by the linear map \(F\). This overlap helps us identify shared characteristics and exploits the structure among the subspaces. It's a powerful tool in figuring out combined properties and reach of linear maps.

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Most popular questions from this chapter

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Sei \(F: V \rightarrow V\) linear mit \(F^{2}=F\). Zeigen Sie, dass es Untervektorr盲ume \(U, W\) von \(V\) gibt mit \(V=U \oplus W\) und \(F(W)=0, F(u)=u\) f眉r alle \(u \in U\).
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