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is the set of all 2x2 upper triangular matrices a vector space? show that it satisfies at least three axioms, or that it fails to satisfy at least one.

User MetalFrog
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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.

User Everzet
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