R help - Nov 2003 - A matrix is full rank is equal to having independent columns?

Dear R listers,

Just a simple question.
If we say an nxm matrix (n>m) is full rank of m,
does this mean that this matrix has linearly independent columns?

They are the same definition or needs some proof?

Thanks for your answer.

Fred

	[[alternative HTML version deleted]]

For any matrix, the following definitions hold:

   row rank: number of linearly independent rows
column rank: number of linearly independent columns

There is a theorem stating that these 2 numbers must be the same 
for any matrix, and (consequently) that number is defined as the 
'rank' of the matrix.

For a matrix which has less columns than rows (as in your example), to say it 
has 'full column rank' would mean that it's rank = number of
columns, and so
yes, by definition all it's columns are linearly independent. I don't
know if
the description 'full rank' has any concrete interpretation for such 
matrices, though.

HTH.

On Monday 03 November 2003 13:32, Feng Zhang wrote:
> Dear R listers,
>
> Just a simple question.
> If we say an nxm matrix (n>m) is full rank of m,
> does this mean that this matrix has linearly independent columns?
>
> They are the same definition or needs some proof?
>
> Thanks for your answer.
>
> Fred
>
> 	[[alternative HTML version deleted]]
>
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help

On 3 Nov 2003 at 13:32, Feng Zhang wrote:

This are the same concept, no additional proof is needed.
The rank of a matrix is the max number of li columns, or rows
(which are the same).

Kjetil Halvorsen
> Dear R listers,
> 
> Just a simple question.
> If we say an nxm matrix (n>m) is full rank of m,
> does this mean that this matrix has linearly independent columns?
> 
> They are the same definition or needs some proof?
> 
> Thanks for your answer.
> 
> Fred
> 
> 	[[alternative HTML version deleted]]
> 
> ______________________________________________
> R-help at stat.math.ethz.ch mailing list
> https://www.stat.math.ethz.ch/mailman/listinfo/r-help

Feng Zhang wrote:
> Dear R listers,
> 
> Just a simple question.
> If we say an nxm matrix (n>m) is full rank of m,
> does this mean that this matrix has linearly independent columns?
> 
Yes.  Now be careful how you define rank.

Cheers

Jason
-- 
Indigo Industrial Controls Ltd.
http://www.indigoindustrial.co.nz
64-21-343-545
jasont at indigoindustrial.co.nz