A problem with basis and dimension

[chefobg57]

Hi!

I am working on the following problem:

If a matrix is antisymmetric (thus A^T = -A), show that

P = {A \in R | A is antisymmetric} is a subset of Rnxn. Also, find the dimension of P.

So far, I've proven that P is a subset of R and I am guessing that, in order for me to find the dimension, I need to find the basis of P first. Here is where I am kind of stuck.

I understand what the properties of the base would be (1. the vectors inside the set will be linearly independent and 2. the basis will be a spanning set for P), but how exactly should I start working so that I can find the basis itself...

A hint would be highly appreciated!

Thanks a bunch guys!

 

[hamster143]

How many numbers do you need to fully describe an arbitrary antisymmetric nxn matrix?

In other words, how many independent components does such matrix have?

 

[chefobg57]

Hm..
Well, I am debating between n and 2n numbers.
The following 2x2 matrix is antisymmetric:
0 a
-a 0

So, it depends on one number, a.

A description of a 3x3 matrix would depend on 3 numbers, a, b, and c and so forth.

Does this mean that this is the answer to the problem? n elements?

Thank you in advance!

 

[hamster143]

Try to write down an asymmetric 4x4 matrix.

 

[chefobg57]

Thanks for pointing that out.

For 2x2 matrices,1 element is necessary.
For 3x3 matrices, 3 elements, for 4x4 - 6 elements, for 5x5 - 9 elements, for 6x6 it's 15 elements..

I am trying to come up with the proper function that will generate such output, given the input (n), but ... I can't really think of anything..

 

[hamster143]

Thanks for pointing that out.

For 2x2 matrices,1 element is necessary.
For 3x3 matrices, 3 elements, for 4x4 - 6 elements, for 5x5 - 9 elements, for 6x6 it's 15 elements..

I am trying to come up with the proper function that will generate such output, given the input (n), but ... I can't really think of anything..

Consider that, say, a 6x6 matrix has 6 elements on the diagonal, 15 elements above the diagonal, and 15 elements below the diagonal. 6+15+15 = 36.

 

[chefobg57]

Oh great!
So can we say then that the dimension for P would be:
(n^2 - n)/2?

Thank you so much!