Math Problem Statement

Let V = ℝ. For u, v ∈ V and a ∈ ℝ define vector addition by u ⊞ v := u + v − 1 and scalar multiplication by a ⊡ u := au − a + 1. Find the following: the sum: 4 ⊞ 5, the scalar multiple: 0 ⊡ 4, the zero vector: 0ₜ, and the additive inverse of x: ⊟x.

Solution

The problem defines a vector space V=RV = \mathbb{R}, and two operations: vector addition u⊞v:=u+v−1u \boxplus v := u + v - 1 and scalar multiplication a⊡u:=au−a+1a \boxdot u := au - a + 1. The task asks for specific results using these operations.

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Math Problem Analysis

Mathematical Concepts

Vector Space
Scalar Multiplication
Vector Addition

Formulas

u ⊞ v := u + v − 1
a ⊡ u := au − a + 1

Theorems

Properties of vector spaces over a scalar field

Suitable Grade Level

Undergraduate (University level)

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