Prove that every linear map on a finite dimensional vector space is continuous

Question

Need to know how to prove that every linear map is continuous. working with finite dimensional vector spaces. It's one of the problems I'm currently working on as revision for an upcoming test.

I know that that a map is continuous if the preimage of any open set is open but can't seem to figure out the full proof.

It's good that you can't figure out the proof, since the claim is not true. — Chris Eagle, Nov 13 '12 at 00:00
@Sigur Yes i am. Chris, why do you claim that it's not true? Thanks — Denis, Nov 13 '12 at 00:06
It's only false in infinite-dimensional vector spaces, Denis. — Kevin Carlson, Nov 13 '12 at 00:10
Thanks @KevinCarlson How would I go about solving it in finite dimensional vector spaces? — Denis, Nov 13 '12 at 00:14

score 5 · Accepted Answer · answered Nov 13 '12 at 00:16

5

Here is the proof for finite dimensional vector spaces:

http://en.wikipedia.org/wiki/Discontinuous_linear_map#If_a_linear_map_is_finite_dimensional.2C_the_linear_map_is_continuous

answered Nov 13 '12 at 00:16

icurays1

17,161

Appreciated! Can't believe I didn't find this all along. Thanks – Denis Nov 13 '12 at 00:21

Prove that every linear map on a finite dimensional vector space is continuous

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