Linear Algebra: Prove that the set of invertible matrices is a Subspace

  • Thread starter Millacol88
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Homework Statement


Is U = {A| A [itex]\in[/itex] nn, A is invertible} a subspace of nn, the space of all nxn matrices?

The Attempt at a Solution


This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and I + (-I) are not, so its not closed under vector addition or scalar multiplication. Is there a way to prove this without assuming the usual vector operations?
 

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Dick
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Homework Statement


Is U = {A| A [itex]\in[/itex] nn, A is invertible} a subspace of nn, the space of all nxn matrices?

The Attempt at a Solution


This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and I + (-I) are not, so its not closed under vector addition or scalar multiplication. Is there a way to prove this without assuming the usual vector operations?
Not until they tell you what the alternative addition and multiplication operations are. I don't think you have to worry about that. Your examples are just fine.
 

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