A problem with basis and dimension

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    Basis Dimension
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Discussion Overview

The discussion revolves around the properties of antisymmetric matrices, specifically focusing on the subset of antisymmetric matrices within the space of nxn matrices. Participants explore how to determine the dimension of this subset and the number of independent components required to describe an arbitrary antisymmetric matrix.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes that to find the dimension of the set of antisymmetric matrices, they need to establish a basis for this set.
  • Another participant questions how many independent components are needed to describe an arbitrary antisymmetric nxn matrix.
  • Some participants suggest that the number of independent components may vary, with one stating that for a 2x2 matrix, 1 element is needed, for a 3x3 matrix, 3 elements, and so forth.
  • There is a suggestion that the dimension for the set of antisymmetric matrices could be represented by the formula (n^2 - n)/2, although this remains unconfirmed.
  • Participants discuss the relationship between the number of elements above the diagonal and the total number of elements in the matrix, noting patterns for different sizes of matrices.

Areas of Agreement / Disagreement

Participants express varying views on the number of independent components required for antisymmetric matrices, with no consensus reached on the exact dimension of the subset P.

Contextual Notes

Some participants mention the need for a function to generate the number of independent components based on the size of the matrix, indicating potential limitations in their current understanding or approach.

Who May Find This Useful

This discussion may be of interest to students and professionals working in linear algebra, matrix theory, or related fields, particularly those exploring properties of matrix structures and dimensions.

chefobg57
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Hi!

I am working on the following problem:

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

P = {A [tex]\in[/tex] 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!
 
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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?
 
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!
 
Try to write down an asymmetric 4x4 matrix.
 
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..
 
chefobg57 said:
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.
 
Oh great!
So can we say then that the dimension for P would be:
(n^2 - n)/2?

Thank you so much!
 

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