We are given a vector space $V = \mathbb{R}^2$ with two binary operations: vector addition $\oplus$ and scalar multiplication $\odot$ as follows:

• Vector addition: $(u_1, v_1) \oplus (u_2, v_2) = (u_1 + u_2 + 3, v_1 + v_2 - 5)$
• Scalar multiplication: $a \odot (u, v) = (au - sa + s, av - ta + t)$

We are tasked with determining the scalars $s$ and $t$ for which $(V, \oplus, \odot)$ is a vector space. To be a vector space, the operations must satisfy several axioms such as:

1. Associativity of addition: $\oplus$ should be associative.
2. Commutativity of addition: $\oplus$ should be commutative.
3. Identity element for addition: There should exist a zero vector $\mathbf{0} = (0,0)$ such that $(u, v) \oplus \mathbf{0} = (u, v)$.
4. Additive inverse: For each $(u, v)$, there should be an element $-(u, v)$ such that $(u, v) \oplus -(u, v) = \mathbf{0}$.
5. Distributivity of scalar multiplication 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 of scalar multiplication over field addition: $(a + b) \odot (u, v) = (a \odot (u, v)) \oplus (b \odot (u, v))$.
7. Compatibility of scalar multiplication: $a \odot (b \odot (u, v)) = (ab) \odot (u, v)$.
8. Identity element of scalar multiplication: $1 \odot (u, v) = (u, v)$.

Step 1: Check the identity element for addition

For the identity element, we require that: $(u, v) \oplus (0, 0) = (u, v)$ Substituting into the definition of $\oplus$: $(u, v) \oplus (0, 0) = (u + 0 + 3, v + 0 - 5) = (u + 3, v - 5)$ For this to equal $(u, v)$, we must have: $u + 3 = u \quad \text{and} \quad v - 5 = v$ This leads to contradictions: $3 = 0 \quad \text{and} \quad -5 = 0$ which are impossible. Thus, there is no identity element for addition unless the operation is modified. Therefore, for no values of $s$ and $t$ can this system form a vector space, since it fails a fundamental requirement.

Would you like to delve into any of the vector space axioms further or explore other aspects of this problem?

Here are 5 related questions you might find helpful:

1. What happens if we change the addition operation to include different constants?
2. How does scalar multiplication affect other vector space properties?
3. Can this space still be useful in other mathematical contexts even if it isn’t a vector space?
4. Could this space form a group instead of a vector space?
5. What are the general criteria for an operation to define a vector space?

Tip: Whenever you are testing if a space is a vector space, always start by checking for the identity element and the distributive properties. If these fail, the space cannot be a vector space.