To find a counterexample for part 1a, let’s examine why this structure fails to be a vector space. We need to check if this modified version of $\mathbb{R}^2$ (with the non-standard addition defined as $\begin{bmatrix} a \\ b \end{bmatrix} + \begin{bmatrix} c \\ d \end{bmatrix} = \begin{bmatrix} a - c \\ b - d \end{bmatrix}$) fails any of the vector space axioms.

One of the critical axioms of a vector space is that addition must be commutative: $\vec{u} + \vec{v} = \vec{v} + \vec{u} \quad \forall \vec{u}, \vec{v} \in \mathbb{R}^2$

With the given definition of addition, however, we can find a counterexample:

1. Select two vectors in $\mathbb{R}^2$, say: $\vec{u} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \quad \text{and} \quad \vec{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$

2. Compute $\vec{u} + \vec{v}$ using the given definition: $\vec{u} + \vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} + \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 1 - 3 \\ 2 - 4 \end{bmatrix} = \begin{bmatrix} -2 \\ -2 \end{bmatrix}$

3. Compute $\vec{v} + \vec{u}$ using the same addition rule: $\vec{v} + \vec{u} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} + \begin{bmatrix} 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 - 1 \\ 4 - 2 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \end{bmatrix}$

4. Observe the result: $\vec{u} + \vec{v} = \begin{bmatrix} -2 \\ -2 \end{bmatrix} \neq \begin{bmatrix} 2 \\ 2 \end{bmatrix} = \vec{v} + \vec{u}$

Since $\vec{u} + \vec{v} \neq \vec{v} + \vec{u}$, this addition operation is not commutative, which violates the axioms required for a vector space. Therefore, this serves as a counterexample to show that the given set with this addition is not a vector space.

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Do you have any questions or would you like more details? Here are some related questions to expand on this topic:

1. What other vector space axioms could be checked for failure in this structure?
2. Could an alternative definition of addition make this a vector space?
3. How would scalar multiplication be affected if a different addition rule is chosen?
4. Can you provide examples of common vector spaces that satisfy all vector space properties?
5. What is the impact of non-standard operations on the structure of mathematical objects like vector spaces?

Tip: When verifying if a set and operations form a vector space, always begin by checking the axioms of vector addition and scalar multiplication, as these often reveal whether the structure can indeed be classified as a vector space.