1. ## Antisymmetric matrix help

Denote the set of 2x2 antisymmetric matrices as S. By making a correspondence between elements of [M 2,2] and four-tuples (a,b,c,d), write the subset of IR^4 corresponding to S as a span. What is the dimension of this subspace?

This question has me stumped! No idea.

Any help is much appreciated,

iExcavate

2. Originally Posted by iExcavate
Denote the set of 2x2 antisymmetric matrices as S. By making a correspondence between elements of [M 2,2] and four-tuples (a,b,c,d), write the subset of IR^4 corresponding to S as a span. What is the dimension of this subspace?

This question has me stumped! No idea.

Any help is much appreciated,

iExcavate

The matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is antisymmetric iff $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}=-\begin{pmatrix}a&c\\b&d\end{pmatrix}=-A^t$ , so then....what?!

Tonio

3. Originally Posted by tonio
The matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ is antisymmetric iff $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}=-\begin{pmatrix}a&c\\b&d\end{pmatrix}=-A^t$ , so then....what?!

Tonio
I know these properties of an antisymmetric matrix, I am just unsure what the question is asking. Do I now make the matrix into 'vector form' by making it (a, c,b,d)?

4. Originally Posted by iExcavate
I know these properties of an antisymmetric matrix, I am just unsure what the question is asking. Do I now make the matrix into 'vector form' by making it (a, c,b,d)?

Yes. Of course, you could also "make it into vector form" as (a,b,c,d), but this is unimportant unless otherwise specified.

Tonio

5. So for the remainder of the question:

"Write the subset of IR^4 corresponding to S as a span. What is the dimension of this subspace?"

What do I do here?

(note IR^4 means R^4)

I don't get it: you said you know the properties of antisymmetric matrices and thus you must know that, with A as in the prior post , it must be that $a=d=0\,,\,\,b=-c\Longrightarrow$ the set of all antisymmetric 2 x 2 matrices can be identified with the set of all vectors $(0,x,-x,0)\in\mathbb{R}^4$ , and it's easy to see this set is actually a subspace of