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 ^_^

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- Oct 30th 2007, 07:09 AMMarianaASubspaces
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 AMPlato
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 AMMarianaA
- Oct 30th 2007, 12:35 PMPlato
You do know that if

is a symmetric matrix then $\displaystyle A = A^T $?**A**

You should know that $\displaystyle \left( {A + B} \right)^T = A^T + B^T \,\& \,\left( {\alpha A} \right)^T = \alpha A^T$. - Oct 30th 2007, 07:39 PMMarianaA
- Oct 31st 2007, 05:04 AMmalweth
all A where $\displaystyle A = A^T$ 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 $\displaystyle (A+B) = (A+B)^T$ and $\displaystyle \alpha \cdot A = \alpha \cdot A^T$ where $\displaystyle A, B \in S_1; \alpha \in \Re$ ? - Oct 31st 2007, 06:17 PMMarianaA