Abstract Algebra: Theory and Applications · Problem 20.11

Fields — Thomas Judson

Mathematics — Abstract Algebra · basis-dimension

Problem

Let

V

be a vector space of dimension

n

. Prove each of the following statements. (a) If

S = \{v_1, \ldots, v_n\}

is a set of linearly independent vectors for

V

, then

S

is a basis for

V

. (b) If

S = \{v_1, \ldots, v_n\}

spans

V

, then

S

is a basis for

V

. (c) If

S = \{v_1, \ldots, v_k\}

is a set of linearly independent vectors for

V

with

k < n

, then there exist vectors

v_{k+1}, \ldots, v_n

such that

\{v_1, \ldots, v_k, v_{k+1}, \ldots, v_n\}

is a basis for

V