MA25C02 | Linear Algebra Example 8: Proving 2x2 Matrices \[\begin{bmatrix} a &1 \\ 1 & b \end{bmatrix}, a, b \in \mathbb{R}\] as Vector Spaces! 📚✨📐 (Reg 2025)

▶ Tutorial Video: Vector Spaces-Example 8

Vector Space Analysis

We are testing the set W consisting of all 2x2 matrices with fixed off-diagonal entries of 1.

The "Rule" for Set W:

a	1
1	b

(Where a and b are any real numbers)

Axiom 1: Closure under Addition

If we add two matrices from the set, the result must stay in the set.

[a, 1; 1, b] + [c, 1; 1, d] = [a+c, 2; 2, b+d]

Result: Since the off-diagonal entries became 2, the result is no longer in W. ❌

Axiom 2 & 3: Algebraic Properties

Properties like Commutativity and Associativity depend on the rules of real numbers.

Result: These hold perfectly because matrix addition is always commutative and associative. ✅

Axiom 4: Additive Identity

A vector space must contain a Zero Vector.

Standard Zero Matrix = [0, 0; 0, 0]

Result: This matrix does not have "1"s on the off-diagonal, so the zero vector is missing from W. ❌

Axiom 6: Closure under Scalar Multiplication

Multiplying a matrix by a scalar k should keep it in the set.

2 × [a, 1; 1, b] = [2a, 2; 2, 2b]

Result: Multiplying by anything other than 1 changes those fixed off-diagonal entries. ❌

FINAL VERDICT: NOT A VECTOR SPACE

Because the set fails to contain the zero vector and is not closed under addition or scalar multiplication, it does not satisfy the requirements of a vector space.