Lenear algebra independance theoretical question

[transgalactic]

there is a space V=Q^4 over Q .
there is a series of vectors (v1 ,v2 ,v3,v4,v5)
that spans V
does the series
(v1 ,v2 ,v3,v4) is a basis of V?
if it is explain
if not give a counter example??

we could have that v5 is a linear combination of others.

??

 

[HallsofIvy]

there is a space V=Q^4 over Q .
there is a series of vectors (v1 ,v2 ,v3,v4,v5)
that spans V
does the series
(v1 ,v2 ,v3,v4) is a basis of V?
if it is explain
if not give a counter example??

we could have that v5 is a linear combination of others.

??

A basis for a vector space has three properties.
1) It spans the space.
2) The vectors in it are linearly independent.
3) The number of vectors in it is the same as the dimension of the vector space.

Further, if any two of those are true, the third must be. You are told that {v1, v2, v3, v4} spans V and you can certainly see that there are four vectors in that set. What is the dimension of V?