Abstract Algebra: Theory and Applications · Problem 20.6

Fields — Thomas Judson

Mathematics — Abstract Algebra · vector-spaces-basis-dimension

Problem

Let

F

be a field and denote the set of

n

-tuples of

F

F^n

. Given vectors

u = (u_1, \ldots, u_n)

and

v = (v_1, \ldots, v_n)

F^n

and

\alpha

F

, define vector addition by

u + v = (u_1, \ldots, u_n) + (v_1, \ldots, v_n) = (u_1 + v_1, \ldots, u_n + v_n)

and scalar multiplication by

\alpha u = \alpha(u_1, \ldots, u_n) = (\alpha u_1, \ldots, \alpha u_n).

Prove that

F^n

is a vector space of dimension

n

under these operations.