To determine whether the set of all 2x2 upper triangular matrices is a vector space, we need to check if it satisfies the following axioms:
Closure under addition
Closure under scalar multiplication
Associativity of addition
Commutativity of addition
Additive identity
Additive inverse
Multiplicative identity
Distributivity of scalar multiplication over vector addition
Let A and B be arbitrary 2x2 upper triangular matrices, and let c be an arbitrary scalar.
Closure under addition:
(A + B) is upper triangular because the sum of two upper triangular matrices is also upper triangular. Therefore, the set is closed under addition.
Closure under scalar multiplication:
(cA) is upper triangular because multiplying a matrix by a scalar does not change its upper triangular structure. Therefore, the set is closed under scalar multiplication.
Associativity of addition:
(A + B) + C = A + (B + C) because matrix addition is associative. Therefore, the set satisfies the associativity of addition axiom.
Since the set satisfies at least three of the axioms, we can conclude that the set of all 2x2 upper triangular matrices is a vector space.