What is the smallest subspace of 3x3 matrices

I want to confirm something:

what is the smallest subspace of 3x3 matrices that contains all symmetric matrices and lower triangular matrices?
- identity(*c)? because that is the only symmetric lower triangular i could think of...

what is the largest subspce that is contained in both of those subspaces?
- identity (*c)?

Answers and Replies

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radou
Homework Helper
If, by 'small subspace', you mean the number of elements in the basis, then the first answer seems right. As for the second one, consider the case when all the diagonal elements are different. What is the dimension of that subspace?

Edit: amof, the first one is not correct. Hint: which element belongs to every subspace? Consider that subspace.

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3? since that subspace is spanned by 3 basis column vectors?
oh, so now there is no restriction on what is subspace spanned by: in first case it had to be 3x3 matrices and now it's just vectors, is this correct to say?

radou
Homework Helper
3? since that subspace is spanned by 3 basis column vectors?
oh, so now there is no restriction on what is subspace spanned by: in first case it had to be 3x3 matrices and now it's just vectors, is this correct to say?
First of all, matrices are vectors, since they are elements of a vector space. Second, an element of a basis is, of course, an element of the vector space considered, so your basis has to consist of 3x3 matrices. Look at the 3x3 null-matrix. Is it symmetric? Is it lower-triangular?

ok i see...
so,for the first one it is zero matrix and identity;
for the second one the answer is still the same... or am I missing something?

radou
Homework Helper
ok i see...
so,for the first one it is zero matrix and identity;
What do you mean by 'zero matrix and identity' ? The trivial subspace consists only of the zero matrix, and has dimension 0.

for the second one the answer is still the same... or am I missing something?
Correct. The basis consists of three matrices.

What do you mean by 'zero matrix and identity' ? The trivial subspace consists only of the zero matrix, and has dimension 0.
it says "ALL", what do I make of that? that's why I included identity even though it says "smallest".... or how should I understand this in english?

Correct. The basis consists of three matrices.
3 3x3?

sorry, I am trying to understand how to look at these problems...

radou
Homework Helper
it says "ALL", what do I make of that? that's why I included identity even though it says "smallest".... or how should I understand this in english?
The trivial substace, consisting of a 3x3 null-matrix, is the smallest subspace of the vector space of all symmetric and lower-triangular 3x3 matrices, since it contains only one element, the 3x3 null-matrix, which satisfies both of your conditions. You should look at the vector space axioms once again. The null-vector (in this case, the null-matrix of order 3) is an element of every vector space. Further on, the trivial subspace is the subspace of every vector space (since it is a subset of every set consisting of the elements of the vector space and since this 'small' subspace is closed under addition and scalar multiplication, which is trivial to show).

StatusX
Homework Helper
I thought you were asking for the smallest subspace of 3x3 matrices consisting of all the 3x3 symmetric and lower-triangular matrices. Every matrix can be written as the sum of a symmetric and lower-triangular matrix, so there is no such proper subspace.

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