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The provided text is a proof in linear algebra that shows that the dimension of the vector space \( W \), denoted as \( \dim(W) = n \), is an even number.

Here's a breakdown of the proof:

1. **Given Information**: We start with a vector space \( W \) where \( \dim(W) = n \).

2. **Rank-Nullity Theorem**: According to the Rank-Nullity Theorem, for a linear map \( f \), we have:
   \[
   \dim(\text{Kern}(f)) + \dim(\text{Bild}(f)) = n
   \]
   where \(\text{Kern}(f)\) is the kernel (null space) of \( f \) and \(\text{Bild}(f)\) is the image (range) of \( f \).

3. **Symmetric Dimension Condition**: The statement says that \(\text{Kern}(f) = \text{Bild}(f)\), implying that the dimensions of the kernel and image are equal. Let \( k = \dim(\text{Kern}(f)) = \dim(\text{Bild}(f)) \).

4. **Equation Derivation**: Substituting into the Rank-Nullity formula, we get:
   \[
   k + k = n \Rightarrow 2k = n
   \]

5. **Conclusion**: Since \( n = 2k \), \( n \) must be an even number.

Therefore, it is proven that the dimension of \( W \), \( n \), is indeed an even number.

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. How does the Rank-Nullity Theorem apply in cases where \(\text{Kern}(f) \neq \text{Bild}(f)\)?
2. What are the implications if \( \dim(W) \) is an odd number?
3. Can this approach be generalized for infinite-dimensional vector spaces?
4. What would happen if \( \dim(\text{Kern}(f)) \neq \dim(\text{Bild}(f)) \) but both are still finite?
5. How would this proof change if \( f \) were an isomorphism?

**Tip**: Remember that the Rank-Nullity Theorem is a powerful tool in linear algebra for understanding the structure of linear transformations.

Show that the dimension of vector space W is even, given dim(W) = n and Kern(f) = Bild(f).

Explore a linear algebra proof that shows the dimension of vector space W is even, using the Rank-Nullity Theorem and properties of Kern(f) and Bild(f). Ideal for undergraduate students studying linear transformations and vector spaces.

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