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Problem 4

Avg difficulty 3.0

Let FF be a field and nn be a positive integer. Define addition and scalar multiplication by

(u1,u2,…,un)+(v1,v2,…,vn)=(u1+v1,u2+v2,…,un+vn)(u_{1},u_{2},\dots,u_{n})+(v_{1},v_{2},\dots,v_{n})=(u_{1}+v_{1},u_{2}+v_{2},\dots,u_{n}+v_{n})

and

α(u1,u2,…,un)=(αu1,αu2,…,αun).\alpha(u_{1},u_{2},\dots,u_{n})=(\alpha u_{1},\alpha u_{2},\dots,\alpha u_{n}).

respectively, for any (u1,u2,…,un),(v1,v2,…,vn)∈Fn(u_{1},u_{2},\dots,u_{n}),(v_{1},v_{2},\dots,v_{n})\in F^{n} and α∈F\alpha\in F. Prove that FnF^{n} is a vector space over FF under the above definitions of addition and scalar multiplication.

Kan, A Complete Introduction to Abstract Algebra, Fields, Vector Spaces, Problem 4

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.