It seems that you are working with a vector space $V = \mathbb{R}^2$, and you've defined a new operation for vector addition, denoted by $\oplus$, which is:

$a \oplus b = \begin{bmatrix} a_1 + a_2 \\ a_2 \cdot b_1 \end{bmatrix}$

where $a = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $b = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ are vectors in $\mathbb{R}^2$.

To clarify, the operation $\oplus$ is defined as:

• The first component of $a \oplus b$ is the regular sum of the first components of $a$ and $b$: $a_1 + a_2$.
• The second component is the product of the second component of $a$ and the first component of $b$: $a_2 \cdot b_1$.

This is a non-standard definition of vector addition, so it would be useful to investigate whether this operation satisfies the properties of vector addition.

Let’s break down the properties that need to be checked for $\oplus$ to define a valid vector addition:

1. Commutativity:

For vector addition to be commutative, we need: $a \oplus b = b \oplus a$ This means: $\begin{bmatrix} a_1 + a_2 \\ a_2 \cdot b_1 \end{bmatrix} = \begin{bmatrix} b_1 + b_2 \\ b_2 \cdot a_1 \end{bmatrix}$ For this equality to hold, the following conditions must be true:

• $a_1 + a_2 = b_1 + b_2$ (first components must add to the same value),
• $a_2 \cdot b_1 = b_2 \cdot a_1$ (second components must satisfy a multiplication equality).

Thus, commutativity depends on these specific conditions.

2. Associativity:

For vector addition to be associative, we need: $(a \oplus b) \oplus c = a \oplus (b \oplus c)$ This requires checking both sides of the equation using the definition of $\oplus$ and seeing if they match. Given the form of the operation, this is likely to be non-trivial to verify directly.

3. Identity Element:

The identity element $e$ should satisfy: $a \oplus e = e \oplus a = a$ We would need to find such a vector $e \in \mathbb{R}^2$, if it exists, that satisfies the equation for all $a$.

4. Inverse Elements:

Each vector $a$ should have an inverse $b$ such that: $a \oplus b = b \oplus a = e$ where $e$ is the identity element. Finding these inverses would be an important part of verifying whether this operation defines a valid vector space.

5. Distributivity of Scalar Multiplication over Vector Addition:

Since you've mentioned scalar multiplication $*$, the distributive property would need to hold: $\lambda \cdot (a \oplus b) = \lambda \cdot a \oplus \lambda \cdot b$ This would require checking how scalar multiplication interacts with your defined vector addition.

Next Steps

You could go ahead and test these properties in more detail to see if this operation satisfies the vector space axioms or if any adjustments are needed.

---

Would you like to check any of these properties in more detail? Or perhaps you have a specific aspect of this operation you want to focus on? Let me know!

Here are some follow-up questions to explore further:

1. How do you think this vector addition will behave under scalar multiplication?
2. Can you find the identity element for this new operation $\oplus$?
3. Do you want to check if this operation satisfies associativity with an example?
4. Would you like me to help you find the inverse of a vector under this new addition?
5. How do you think this non-standard addition compares with the usual vector addition in $\mathbb{R}^2$?

Tip: When testing new operations like this, always begin by checking the core vector space properties: commutativity, associativity, identity, and inverses.