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## Main Question or Discussion Point

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

any hits? kinda confused with this span thing.

any hits? kinda confused with this span thing.

- Thread starter 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.

any hits? kinda confused with this span thing.

matt grime

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

mathwonk

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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

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Yooyo, back to you! Are you allowed to use the fact that R

mathwonk

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can you prove one vector cannot spane R^2?

mathwonk

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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.

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