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

[Millacol88]

Homework Statement

Is U = {A| A $\in$ ⁿℝⁿ, A is invertible} a subspace of ⁿℝⁿ, 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?

 

[Dick]

Homework Statement

Is U = {A| A $\in$ ⁿℝⁿ, A is invertible} a subspace of ⁿℝⁿ, 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.