Help With Algebraic Extensions/Bases

[audiowize]

Homework Statement

Determine the basis for {n x n matrices with rational elements}

Determine the basis for {a+b*3^.5 : a,b in Q(3^.5 }

The Attempt at a Solution

I am not super sure about the basis for an n x n matrix. I know the dimension will be n-squared, and how the elements will look, but actually sitting and writing out what the basis is seems tricky, or I am missing something...

On the second question, I think the basis is {1,3^.5}, but I would love some confirmation...

Ah, the joys of Memorial Day homework! At least I have until Wednesday...

Thanks in advance!

 

[mighty2000]

Thank you for your question. The basis for {n x n matrices with rational elements} is the set of n x n matrices with 1 in the upper left corner and 0's everywhere else. This is known as the standard basis for n x n matrices.

For the second question, the basis for {a+b*3.5 : a,b in Q(3.5)} is indeed {1, 3.5}. This is because any element in this set can be written as a linear combination of these two basis vectors.

I hope this helps with your homework. Happy Memorial Day!