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

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In summary, the conversation discusses the concept of spanning sets in linear algebra, specifically in the context of n vectors in Rm. The participants consider whether a set of n vectors can span all of Rm and discuss the definition of dimension and how it relates to spanning sets. They also mention the use of quotient spaces and provide a quick inductive argument for proving that one vector cannot span R^2.
  • #1
yooyo
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Could a set of n verctors in Rm span all of Rm when n<m?
any hits? kinda confused with this span thing.:confused:
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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?
 
  • #5
can you prove one vector cannot spane R^2?
 
  • #6
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.
 

1. What is the definition of "span" in linear algebra?

The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be created by multiplying each vector by a scalar and adding them together.

2. How is the dimensionality of a vector space related to its span?

The dimensionality of a vector space is the number of vectors needed to span the space. In other words, it is the minimum number of vectors required to create every possible vector in the space through linear combinations.

3. Can a set of n vectors in Rm span all of Rm when n>m?

No, a set of n vectors in Rm cannot span all of Rm when n>m. This is because there are more dimensions in Rm than there are vectors, so there will always be some vectors that cannot be created through linear combinations of the given vectors.

4. Can a set of n vectors in Rm span all of Rm when n=m?

Yes, a set of n vectors in Rm can span all of Rm when n=m. This is because there are exactly enough vectors to create every possible vector in the space through linear combinations.

5. Can a set of n vectors in Rm span all of Rm when n

Yes, a set of n vectors in Rm can span all of Rm when n

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