Determine whether the subset S of M2x2 (2x2 are the subscripts for M,

[pyroknife]

Determine whether the subset S of M2x2 (2x2 are the subscripts for M, idk how to do put it on here) is a subspace.

Let S be the set of all diagonal matrices.

To check if something is a subspace, my teacher gave us 3 conditions.

1.) 0 vector is in S
2) if U and V are in S, then U+V is in S
3) If V is in S and c is a scalar, then cV is in S.

I'm not really sure how to check the first condition. A guess no vectors on a diagnol 2x2 matrix can be the 0 vector,thus S is not a subspace in this case?

 

[Dick]

The zero matrix is definitely diagonal. The matrices in this question are your 'vectors'.

 

[pyroknife]

Wait how is a zero matrix diagonal? My prof said "diagonal means square, with nonzero entries only appearing on the diagonal." I missed the "only" part, I think that means the diagonal can still be 0 or nonzero, but no elements off the diagonal can be nonzero.But if so.
then the 1st condition is satisfied.

2. Condition 2 implies that if we have a U and V in S, then U+V must also be diagonal. Which is true.
3. Condition 3 implies if we have a V in S, then cV is also diagnol, which is true.
Thus S is a subspace.
Is this correct?

 

[Dick]

Wait how is a zero matrix diagonal? My prof said "diagonal means square, with nonzero entries only appearing on the diagonal."


But if so.
then the 1st condition is satisfied.

2. Condition 2 implies that if we have a U and V in S, then U+V must also be diagonal. Which is true.
3. Condition 3 implies if we have a V in S, then cV is also diagnol, which is true.
Thus S is a subspace.
Is this correct?

Yes, 'nonzero entries only appearing on the diagonal' only means the off-diagonal elements have to be zero. It doesn't say the diagonal elements have to be zero.

 

[pyroknife]

So this matrix is a subspace. If the way I explained why it is sufficient or do I need to write out arbitrary matrices and demonstrate each 3 conditoins?

 

[Dick]

So this matrix is a subspace. If the way I explained why it is sufficient or do I need to write out arbitrary matrices and demonstrate each 3 conditoins?

You didn't really explain anything. You just said it was true. I assumed you knew what you were talking about. Your teacher might not. Yes, write out the form of a diagonal matrix and show each one is true.