Question

Let V = (2, [infinity]). For u, v in V and a in R, define vector addition by u ⊕ v = uv - 2(u + v) + 6 and scalar multiplication by a * u = (u - 2)^a + 2.

Answer 1

The given definitions define vector addition and scalar multiplication for vectors in V = (2, ∞) and scalars in R.

Step-by-step explanation:

Vector Addition (u ⊕ v):

Component-wise multiplication: Each component of u and v is multiplied.

Double subtraction: The sum of both u and v is subtracted twice from the product (2(u + v)).

Constant addition: Finally, 6 is added to the result.

Scalar Multiplication (a * u):

Exponential term: Each component of u is reduced by 2 and then raised to the power of the scalar a.

Constant addition: Finally, 2 is added to the result of the exponential term for each component.

These definitions allow for performing addition and scalar multiplication on vectors in V using the specified operations on their components. They capture some non-standard properties compared to common vector space operations, making it an interesting variation on typical vector operations.