The document contains exercises from a Linear Algebra course. Here's an overview and detailed instructions for solving the problems:

Summary of Exercises

Aufgabe 8-1

1. Part (a): Prove that $U_1 + U_2 + U_3 = \mathbb{K}^3$, where $U_1, U_2, U_3$ are given subspaces of $\mathbb{K}^3$.
2. Part (b): Demonstrate that the sum $U_1 + U_2 + U_3$ is not direct by finding a vector with two distinct representations.
3. Part (c): For the subspace $U \subseteq \mathbb{K}^4$, find a complementary subspace $W \subseteq \mathbb{K}^4$ such that $\mathbb{K}^4 = U \oplus W$, and prove it.

Aufgabe 8-2

1. Part (a): Provide and prove an example of a subset $U \subseteq \mathbb{R}^2$ that satisfies closure under addition but is not a subvector space.
2. Part (b): Provide and prove an example of a subset $U \subseteq \mathbb{R}^2$ that satisfies closure under scalar multiplication but is not a subvector space.

Aufgabe 8-3

1. Part (a): Prove that the space of $m \times n$ matrices over a field $K$, $M(m \times n, K)$, is a vector space.
2. Part (b): Prove that the set of symmetric $n \times n$ matrices over $K$, $\text{Sym}(n, K)$, forms a subvector space.
3. Part (c): Prove properties of symmetric ($A_{\text{sym}}$) and skew-symmetric ($A_{\text{alt}}$) matrices derived from a decomposition of matrices in $M(n \times n, \mathbb{R})$.
4. Part (d): Show the direct sum decomposition $M(n \times n, \mathbb{R}) = \text{Sym}(n, \mathbb{R}) \oplus \text{Alt}(n, \mathbb{R})$.

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Next Steps

I can help solve these problems step by step. Do you want detailed solutions for any specific parts of the exercises, or an overview of the approach to solve them?

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Follow-Up Questions

1. Would you like a breakdown of vector space proofs for specific tasks?
2. Are you interested in examples for subsets that aren't subspaces (Aufgabe 8-2)?
3. Should we dive into matrix operations and their properties for Aufgabe 8-3?
4. Do you need a refresher on direct sums and their applications in Aufgabe 8-1 and 8-3?
5. Would you like a guide on how to prove closure under addition and scalar multiplication?

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Tip: Always verify the closure properties and zero vector presence when working with subspaces. These are common points of confusion.