We are given the set $V = \mathbb{R}^2$ along with two binary operations on $V$, namely vector addition $\oplus$ and scalar multiplication $\odot$, defined as follows:

For vector addition: $(u_1, v_1) \oplus (u_2, v_2) = (u_1 + u_2 + 3, v_1 + v_2 - 5)$ For scalar multiplication: $a \odot (u, v) = (au - sa + s, av - ta + t)$ where $s$ and $t$ are some scalar values to be determined. We are asked to find $s$ and $t$ such that $(V, \oplus, \odot)$ forms a vector space.

Conditions for a Vector Space

For $(V, \oplus, \odot)$ to form a vector space, the following conditions must hold:

1. Associativity of Addition: For all $(u_1, v_1), (u_2, v_2), (u_3, v_3) \in V$, we must have: $((u_1, v_1) \oplus (u_2, v_2)) \oplus (u_3, v_3) = (u_1, v_1) \oplus ((u_2, v_2) \oplus (u_3, v_3)).$

2. Commutativity of Addition: For all $(u_1, v_1), (u_2, v_2) \in V$, we must have: $(u_1, v_1) \oplus (u_2, v_2) = (u_2, v_2) \oplus (u_1, v_1).$

3. Additive Identity: There must be an element $(0, 0) \in V$ such that for all $(u, v) \in V$: $(u, v) \oplus (0, 0) = (u, v).$

4. Additive Inverse: For each $(u, v) \in V$, there must be an element $(-u, -v) \in V$ such that: $(u, v) \oplus (-u, -v) = (0, 0).$

5. Distributivity of Scalar Multiplication: Scalar multiplication should distribute over vector addition: $a \odot ((u_1, v_1) \oplus (u_2, v_2)) = (a \odot (u_1, v_1)) \oplus (a \odot (u_2, v_2)).$

6. Distributivity over Scalars: For all $a, b \in \mathbb{R}$ and $(u, v) \in V$, we must have: $(a + b) \odot (u, v) = (a \odot (u, v)) \oplus (b \odot (u, v)).$

7. Scalar Identity: For the scalar $1 \in \mathbb{R}$ and any $(u, v) \in V$, we must have: $1 \odot (u, v) = (u, v).$

Key Conditions to Focus On

The commutativity of addition and the identity conditions for addition and scalar multiplication are often the most restrictive, so we'll start by testing those to determine values for $s$ and $t$.

Let's begin by solving for $s$ and $t$.It appears that there are no nontrivial solutions for $s$ and $t$ based solely on the commutativity of addition and scalar identity conditions. This suggests that the given operations for $\oplus$ and $\odot$ might not lead to a valid vector space structure for any choice of $s$ and $t$.

In summary, it seems that the operations defined for vector addition and scalar multiplication violate the basic axioms of a vector space, particularly the commutativity of addition and the scalar identity condition. Therefore, no values of $s$ and $t$ will allow $(V, \oplus, \odot)$ to form a valid vector space.

Do you have any specific parts you'd like to explore further, or would you like a more detailed explanation of vector space axioms?

Here are some related questions you could consider:

1. How is scalar multiplication typically defined in vector spaces?
2. What are the key properties of vector addition in vector spaces?
3. Why is the identity element important in vector spaces?
4. What would happen if we changed the definitions of $\oplus$ or $\odot$?
5. How do distributivity laws function in vector spaces?

Tip: When determining if a set with operations forms a vector space, always test basic axioms like commutativity and scalar identity first—they're often the quickest indicators of potential issues.