221: Set of Vectors Space Homomorphisms Constitutes Vectors Space|T.B.P.

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A description/proof of that set of vectors space homomorphisms constitutes vectors space

Topics

About: vectors space

The reader will have a description and a proof of the proposition that for any 2 vectors spaces over any same field, the set of vectors space homomorphisms constitutes a vectors space.

For any vectors spaces,

V_{1}, V_{2}

, over any same field,

F

, the set of vectors space homomorphisms,

H o m (V_{1}, V_{2}) := {f : V_{1} \to V_{2}}

, is a vectors space over

F

with the vectors space operation defined as

(r_{1} f_{1} + r_{2} f_{2}) (v) = r_{1} f_{1} (v) + r_{2} f_{2} (v)

where

r_{i} \in F

f_{i} \in H o m (V_{1}, V_{2})

, and

v \in V_{1}

r_{1} f_{1} (v) + r_{2} f_{2} (v) \in V_{2}

. So,

r_{1} f_{1} + r_{2} f_{2}

is a map from

V_{1}

V_{2}

. Is

r_{1} f_{1} + r_{2} f_{2}

a vectors space homomorphism? For any

r_{3}, r_{4} \in F

and any

v_{1}, v_{2} \in V_{1}

(r_{1} f_{1} + r_{2} f_{2}) (r_{3} v_{1} + r_{4} v_{2}) = r_{1} f_{1} (r_{3} v_{1} + r_{4} v_{2}) + r_{2} f_{2} (r_{3} v_{1} + r_{4} v_{2}) = r_{1} r_{3} f_{1} (v_{1}) + r_{1} r_{4} f_{1} (v_{2}) + r_{2} r_{3} f_{2} (v_{1}) + r_{2} r_{4} f_{2} (v_{2}) = r_{3} (r_{1} f_{1} (v_{1}) + r_{2} f_{2} (v_{1})) + r_{4} (r_{1} f_{1} (v_{2}) + r_{2} f_{2} (v_{2})) = r_{3} ((r_{1} f_{1} + r_{2} f_{2}) (v_{1})) + r_{4} ((r_{1} f_{1} + r_{2} f_{2}) (v_{2}))

. So, yes,

r_{1} f_{1} + r_{2} f_{2}

is a vectors space homomorphism.

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