Does adding vectors to a basis in R^n always result in a dependent set?

[nhrock3]

A={a1,a2,..ak}
is a group of vectors which is dependent on R^n k>=2

1.
if a1 is a linear combination of a2..ak then
a2..ak is independent
?
i think it doesn't because inside a2..ak we could have another vector which is dependent on the others

2.if k>n then A create R^n

i think it does because the numbe of vectors is bigger then n
so i will create R^n

 

[radou]

for 1. yes, that's true.

for 2. you probably meant, do they "span" R^n. In this case, look at R^3, for example, and take the set {(1, 0 ,0), (2, 0, 0), (3, 0, 0), (4, 0, 0)}. k > n, and do they span R^3?

 

[nhrock3]

thanks
so we need to demand independent group of vectors
?

 

[radou]

If your number of vectors is greater than the dimension of the space, they can't possibly be independent.

 

[nhrock3]

but inside this big k group could be "n" vectors which are independent
and they will span R^n

 

[radou]

Of course, spanning is another issue.

For example, if you take any basis for R^n, and add some vectors to it, the resulting set of vectors will be dependent, and it will still span R^n.