Subspaces

**MarianaA** · October 30th 2007, 07:09 AM

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 ^_^

**Plato** · October 30th 2007, 07:14 AM

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?

**MarianaA** · October 30th 2007, 09:50 AM

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.

**Plato** · October 30th 2007, 12:35 PM

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

**MarianaA** · October 30th 2007, 07:39 PM

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

**malweth** · October 31st 2007, 05:04 AM

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 $A, B \in S_1; \alpha \in \Re$ ?

**MarianaA** · October 31st 2007, 06:17 PM

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

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