# [R]A matrix is full rank is equal to having independent columns?

Deepayan Sarkar deepayan at stat.wisc.edu
Mon Nov 3 20:56:49 CET 2003

```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?
>
>
> Fred
>
> 	[[alternative HTML version deleted]]
>
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```