Could a set of n verctors in Rm span all of Rm when n<m?

[yooyo]

Could a set of n verctors in Rm span all of Rm when n<m?
any hits? kinda confused with this span thing.

 

[matt grime]

R^m is m dimensional real space (it is easy to write down m independent vectors that span).

Just look at the definitions: the dimension is the minimal cardinality of a spanning set.

 

[mathwonk]

you still have to prove that less than nvectors cannot span R^n.

i.e. you have to prove that the space of n tuples of real numbers has dimension n.

look at my web notes on linear algebra.

 

[HallsofIvy]

Unfortunately, Yooyo did not give any indication as to what he had tried and so we have no idea what facts he can use!

Yooyo, back to you! Are you allowed to use the fact that Rⁿ has dimension n or is proving that part of your problem?

 

[mathwonk]

can you prove one vector cannot spane R^2?

 

[mathwonk]

here is a quick inductive argument, if you know about quotient spaces.

case 1, there is no linear surjection from R1 to any higher dimensional space.

if there is a linear surjection from Rn to Rm, where n <m, then the composite surjection from Rn to Rm/em = Rm-1 is not injective.

hence there is a lineaer surjection from some subspace Rn-1 to Rm-1, impossible by inductive hypothesis.