# Math Help - Matrix/basis question

1. ## Matrix/basis question

I can see that this is true but I am having trouble presenting a convincing argument.
Let V = $M_{n\times n} (\mathbb{R})$. Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.

The second part of the question however I have no idea where to start:
Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.

2. Originally Posted by worc3247
I can see that this is true but I am having trouble presenting a convincing argument.
Let V = $M_{n\times n} (\mathbb{R})$. Show that V has a basis with the property that each matrix in the basis is either symmetric or skew symmetric.
If:

$F_1=\left\{{A\in{V}:A^t=A}\right\},\quad F_2=\left\{{A\in{V}:A^t=-A}\right\}$

then,

$V=F_1\oplus F_2$

so, the union of a basis of $F_1$ with a basis of $F_2$ is a basis of $V$.

Fernando Revilla

3. Originally Posted by worc3247
Let V = $M_{n\times n} (\mathbb{R})$.

...

I have no idea where to start:
Show also that V has a basis with the property that each matrix in the basis is an invertible matrix.
Start with a basis for V in which the first element in the basis is the identity matrix $I$. You can add a multiple of $I$ to each other element in the basis so as to make that element invertible, and the resulting set of matrices will still be a basis.