Originally posted by lethe
oh and i assume that you know what it means for vectors to be linearly independent,

Vectors
v_{i} (i=1,2,3,...) in R
^{n} are
independent iff
a
_{1}v_{1}+a
_{2}v_{2}+a
_{3}v_{3}+...=0
implies that
a
_{1}+a
_{2}+a
_{3}+...=0
and what a basis of a vector space is.

A set of vectors (
v_{1},
v_{2},
v_{3},...) is a
basis for a vector space V iff
1.
v_{1},
v_{2},
v_{3},...
span^{*} V.
2.
v_{1},
v_{2},
v_{3},... are independent.
^{*}span(
v_{1},
v_{2},
v_{3},...)={a
_{1}v_{1}+a
_{2}v_{2}+a
_{3}v_{3}+...for a
_{i} in R}
edit: fixed subscript bracket