Subspaces

• Oct 30th 2007, 07:09 AM
MarianaA
Subspaces
Hi I have two questions, I hope some one can help me.

Which subsets of the vector space Mnn are subspaces?
a) The set of all n x n symmetric matrices.
b) The set of all n x n diagonal matrices.

Thanks ^_^
• Oct 30th 2007, 07:14 AM
Plato
Once again all one must is to check two conditions:
1) is the set closed with respect to addition?
2) is the set closed with respect to scalar multiplication?
• Oct 30th 2007, 09:50 AM
MarianaA
Quote:

Originally Posted by Plato
Once again all one must is to check two conditions:
1) is the set closed with respect to addition?
2) is the set closed with respect to scalar multiplication?

Oh ok, can you give me a hint on how I can start it. :(
• Oct 30th 2007, 12:35 PM
Plato
You do know that if A is a symmetric matrix then $A = A^T$?
You should know that $\left( {A + B} \right)^T = A^T + B^T \,\& \,\left( {\alpha A} \right)^T = \alpha A^T$.
• Oct 30th 2007, 07:39 PM
MarianaA
Quote:

Originally Posted by Plato
You do know that if A is a symmetric matrix then $A = A^T$?
You should know that $\left( {A + B} \right)^T = A^T + B^T \,\& \,\left( {\alpha A} \right)^T = \alpha A^T$.

Yeah I know that the transpose of A = A.
But How I can imply that to the subsets
• Oct 31st 2007, 05:04 AM
malweth
all A where $A = A^T$ is the definition of your first subset. From there you need to prove that the summation of two elements from the subset results in an element of your subset and that any scalar multiplication also has the same result (definition of a subspace).

e.g. Can you prove that $(A+B) = (A+B)^T$ and $\alpha \cdot A = \alpha \cdot A^T$ where $A, B \in S_1; \alpha \in \Re$ ?
• Oct 31st 2007, 06:17 PM
MarianaA
Quote:

Originally Posted by malweth
all A where $A = A^T$ is the definition of your first subset. From there you need to prove that the summation of two elements from the subset results in an element of your subset and that any scalar multiplication also has the same result (definition of a subspace).

e.g. Can you prove that $(A+B) = (A+B)^T$ and $\alpha \cdot A = \alpha \cdot A^T$ where $A, B \in S_1; \alpha \in \Re$ ?

I got it.
Thanks (Handshake)