To determine if $\mathbb{R}^2$ with the given operations is a vector space, we must check if it satisfies the axioms of a vector space under the defined operations.

1. Addition: Defined as $[a; b] + [c; d] = [a - c; b - d]$.
2. Scalar Multiplication: Defined as usual, so for a scalar $k$ and a vector $[a; b]$, we have $k \cdot [a; b] = [k \cdot a; k \cdot b]$.

Let's go through the key properties a vector space must satisfy, focusing on addition and scalar multiplication:

1. Closure under addition: For any two vectors $[a; b]$ and $[c; d]$, their sum is $[a - c; b - d]$. Since subtraction is closed in $\mathbb{R}^2$, this is satisfied.

2. Associativity of addition: We need $( [a; b] + [c; d] ) + [e; f] = [a; b] + ( [c; d] + [e; f] )$. However, given that addition is defined as $[a; b] + [c; d] = [a - c; b - d]$, associativity may not hold as expected in usual vector spaces due to the unusual subtraction operation.

3. Existence of an additive identity: In standard vector spaces, the zero vector $[0; 0]$ serves as the additive identity. But with this unusual addition, there’s no $[0; 0]$ that works as an identity in the traditional sense since $[a; b] + [0; 0] = [a; b] - [0; 0] = [a; b]$ would not yield an identity under subtraction.

4. Existence of an additive inverse: We require each vector $[a; b]$ to have an inverse $[c; d]$ such that $[a; b] + [c; d] = [0; 0]$. This is complicated by the fact that addition is defined as subtraction.

Given these issues, particularly with associativity and identity, $\mathbb{R}^2$ under these operations does not satisfy the axioms of a vector space.

Would you like a deeper dive into any specific vector space axiom or clarification on why this structure fails?

Here are some related questions for further exploration:

1. How does the requirement for an additive identity impact vector space structure?
2. Can non-standard addition operations create valid vector spaces?
3. What properties must scalar multiplication satisfy in vector spaces?
4. Are there examples of non-standard vector spaces in other mathematical fields?
5. What happens if we change the addition operation to be traditional addition?

Tip: Always verify the existence of an additive identity and inverses when assessing vector space validity.