Find Basis for diagonal matrix

[Clandry]

I'm not sure how to start this problem.
All i know is a diagonal matrix consists of all 0 elements except along the main diagonal.

But how do I even find a basis for this?

 

[haruspex]

What would a basis look like? It would be set of nxn matrices such that... you can do what with them?

 

[Clandry]

What would a basis look like? It would be set of nxn matrices such that... you can do what with them?

For this case, a basis consists of all matrices such that all nxn diagonal matrices can be written as a linear combination of them?

 

[haruspex]

For this case, a basis consists of all matrices such that all nxn diagonal matrices can be written as a linear combination of them?

Yes. What's the simplest matrix you can think of that might be useful in creating such a basis?

 

[Clandry]

Yes. What's the simplest matrix you can think of that might be useful in creating such a basis?

This is where I get stuck. I've only been taught and done problems where the basis is a set of "vectors."

I saw somewhere that the basis for a 2x2matrix is
1 0
0 0

0 1
0 0

0 0
1 0

0 0
0 1


if it were a 2x2 diagonal would it be
1 0
0 0

and
0 0
0 1
?

 

[haruspex]

Ok. Now try 3x3.

 

[Clandry]

Ok. Now try 3x3.

1 0 0
0 0 0
0 0 0

0 0 0
0 1 0
0 0 0

0 0 0
0 0 0
0 0 1if it's an nxn matrix, wouldn't that give an infinite amount of matrices for the bases?The answer in the back of the book is
37. B = {eii | 1 ≤ i ≤ n} the "ii" part is supposed to be subscripts for e. I'm bad at interpreting these kind of answers, what is it saying?

 

[haruspex]

if it's an nxn matrix, wouldn't that give an infinite amount of matrices for the bases?

You had 2 for 2x2 and 3 for 3x3. Why would you get infinitely many for nxn?

 

[Clandry]

You had 2 for 2x2 and 3 for 3x3. Why would you get infinitely many for nxn?

oops i mean n amount.

 

[haruspex]

So you have the answer.
The e_ii notation used in the book apparently means the nxn matrix that has 1 at the (i, i) position and 0 everywhere else. I don't know how standard that is. Should be defined in the book somewhere.