Let's go through each part of this problem step-by-step:

1. Defining a Linear Map $f: V \to V$ with $\ker(f) = \operatorname{Bild}(f)$:

   • To define a linear map $f$ from $V$ to $V$ with $\ker(f) = \operatorname{Bild}(f)$, we need to think about an endomorphism of the vector space $V$ with this property.
   • Since $V$ is spanned by the elements $\{1, T^2, T^4, T^6\}$, any linear map $f$ can be represented by a matrix with respect to this basis.
   • A possible choice for such an $f$ is to create a map that "shifts" elements in a way that reduces dimension by mapping certain elements to others in the span, satisfying $\ker(f) = \operatorname{Bild}(f)$.

2. Choosing a Basis $\mathcal{B}$ of $V$ and Determining $\mathcal{B} M_{\mathcal{B}}(f)$:

   • The standard basis $\mathcal{B} = \{1, T^2, T^4, T^6\}$ can be used.
   • Once we define $f$, we can represent it as a matrix $M_{\mathcal{B}}(f)$ in this basis.
   • To find $M_{\mathcal{B}}(f)$, we apply $f$ to each basis vector and express the results as linear combinations of the basis vectors. This will help in writing out the matrix of $f$.

3. Proof: If $W$ is a Finite-Dimensional $K$-Vector Space and $f: W \to W$ is a Linear Map with $\ker(f) = \operatorname{Bild}(f)$, then $\dim(W)$ is Even.

   • This result can be shown using the Rank-Nullity Theorem, which states $\dim(W) = \operatorname{rank}(f) + \operatorname{nullity}(f)$.
   • Since $\ker(f) = \operatorname{Bild}(f)$, we have $\operatorname{rank}(f) = \operatorname{nullity}(f)$.
   • Let $\operatorname{rank}(f) = k$. Then, $\dim(W) = k + k = 2k$, which is even.
   • This proves that $\dim(W)$ must indeed be an even number.

Would you like further details on each part, or do you have specific questions? Here are some related questions to explore:

1. What are examples of linear maps with $\ker(f) = \operatorname{Bild}(f)$?
2. How can we construct matrices for linear maps in polynomial vector spaces?
3. How does the Rank-Nullity Theorem apply to this type of linear transformation?
4. What does it mean for a space to have an even dimension?
5. How do we verify the kernel and image equality for a given transformation?

Tip: When constructing a matrix for a linear map, always consider the action of the map on each basis vector and express the results in terms of the basis. This approach will guide you in writing the matrix representation correctly.