Linear Algebra: Vector Space Example 2 📚✨💡 | (1,x1)+(1,x2) = ( 1, x1+x2), k(1,x)=(1,kx)

Problem: Vector Space Verification

Tutorial Video

Check whether the set of all pairs of real numbers of the form \((1, x)\) with the operations \((1, x_1) + (1, x_2) = (1, x_1 + x_2)\) and \(k(1, x) = (1, kx)\) is a vector space.

Step-by-Step Solution

Let \(V = \{(1, y) \mid y \in \mathbb{R}\}\). We check the 10 axioms of a vector space:

✓ Satisfied 1. Closure under Addition For any \((1, y), (1, y') \in V\), the sum \((1, y + y')\) maintains the form \((1, \text{real})\).

✓ Satisfied 2. Commutativity of Addition \[ (1, y) + (1, y') = (1, y + y') = (1, y' + y) = (1, y') + (1, y) \]

✓ Satisfied 3. Associativity of Addition \[ ((1, y) + (1, y')) + (1, y'') = (1, y + y' + y'') = (1, y) + ((1, y') + (1, y'')) \]

✓ Satisfied 4. Additive Identity The identity is \((1, 0)\), since \((1, y) + (1, 0) = (1, y + 0) = (1, y)\).

✓ Satisfied 5. Additive Inverse For every \((1, y)\), the inverse is \((1, -y)\), since \((1, y) + (1, -y) = (1, 0)\).

✓ Satisfied 6. Closure under Scalar Multiplication For \(k \in \mathbb{R}\), \(k(1, y) = (1, ky)\), which is still an element of \(V\).

✓ Satisfied 7. Distributivity (Vector Addition) \[ k((1, y) + (1, y')) = k(1, y + y') = (1, k(y + y')) = (1, ky + ky') \] \[ k(1, y) + k(1, y') = (1, ky) + (1, ky') = (1, ky + ky') \]

✓ Satisfied 8. Distributivity (Scalar Addition) \[ (k + \ell)(1, y) = (1, (k + \ell)y) = (1, ky + \ell y) \] \[ k(1, y) + \ell(1, y) = (1, ky) + (1, \ell y) = (1, ky + \ell y) \]

✓ Satisfied 9. Scalar Identity \[ 1(1, y) = (1, 1y) = (1, y) \]

Conclusion: All axioms are satisfied. The set \(V\) is a vector space over \(\mathbb{R}\).