Linear Algebra: Vector Space Example 4📚✨💡 | x+y=xy and cx=x^c

▶ Tutorial Video: Vector Spaces-Example 4

SOLVED EXAMPLE: Non-Standard Vector Space

Let \(V\) be the set of all positive real numbers. Define the vector addition and scalar multiplication as follows:

\[ x + y = xy \quad \text{and} \quad \alpha x = x^\alpha \]

Determine whether or not \(V\) is a vector space over \(\mathbb{R}\) with respect to the above operations, where \(\alpha\) is a real number.

Axiomatic Verification

1. Closure under Addition \( u + v = uv \). Since \( u > 0 \) and \( v > 0 \), their product \( uv > 0 \). Thus \( uv \in V \). ✓

2. Commutativity \( u + v = uv = vu = v + u \). ✓

3. Associativity \( (u + v) + w = (uv)w = u(vw) = u + (v + w) \). ✓

4. Additive Identity (Zero Vector) We need \(\mathbf{0} \in V\) such that \( u + \mathbf{0} = u \).
\( u \cdot \mathbf{0} = u \implies \mathbf{0} = 1 \). Since \( 1 > 0 \), the zero vector is \( 1 \). ✓

5. Additive Inverse For each \( u \in V \), we need \(-u\) such that \( u + (-u) = 1 \).
\( u \cdot (-u) = 1 \implies -u = \frac{1}{u} \). Since \( u > 0 \), \( \frac{1}{u} \in V \). ✓

6. Closure under Scalar Multiplication \( c u = u^c \). For any \( u > 0 \) and \( c \in \mathbb{R} \), \( u^c \) is always a positive real number. Thus \( u^c \in V \). ✓

7. Distributivity (Scalar over Vectors) \( c(u + v) = (uv)^c = u^c v^c \) and \( cu + cv = u^c \cdot v^c \). Both sides are equal. ✓

8. Distributivity (Vector over Scalars) \( (c + d)u = u^{c+d} = u^c u^d \) and \( cu + du = u^c \cdot u^d \). Both sides match. ✓

9. Compatibility of Scalars \( c(du) = c(u^d) = (u^d)^c = u^{cd} = (cd)u \). ✓

10. Scalar Identity \( 1u = u^1 = u \). ✓

Conclusion: All ten vector space axioms are satisfied. Therefore, the set of positive real numbers \(V\) under these non-standard operations is indeed a Vector Space over \(\mathbb{R}\).