Linear Independence Question #2

Jan 2017
209
3
Toronto
Am I correct that if a matrix \(\displaystyle A_{m ~X~ n} \) (m < n) and its rank is m. Since it has some free variables. By definition the matrix A is linear dependence.

Please review the following matrix A
1.000 -1.000 1.000 5.000
-2.000 3.000 1.000 2.000
3.000 4.000 -2.000 3.000

My program keeps telling me A is a row linear independence matrix, which appears to make sense because each row is not a combination of the others.

However, according to the above definition, the matrix A contains free variables and therefore it must be linear dependence.

So... is A linear dependence or independence?
 
Last edited by a moderator:
Jan 2016
93
48
Athens, OH
I'm not familiar with the term "linear dependence" for a matrix. Given an m by n matrix of row rank m (equal to column rank = rank), the m rows (n component vectors) are linearly independent. If m<n, the n columns are linearly dependent (the situation n<m is impossible for rank m), and if m=n the n columns are linearly independent.

I don't understand what you mean by saying "A has free variables".
 
Jan 2017
209
3
Toronto
free variable = non pivot column

In the above example:

non-pivot column
===========
5
2
3
 
Jan 2017
209
3
Toronto
I got it. The matrix A is linear dependent because it contains one free variable.
 

skipjack

Forum Staff
Dec 2006
21,387
2,410
Did the formal definitions that you were taught use the word "dependence" or "independence", or did they just use "dependent" or "independent"?
 

skipjack

Forum Staff
Dec 2006
21,387
2,410
Okay. Some people use "linear dependent" and "linear independent", but I prefer "linearly dependent" and "linearly independent" on grammatical grounds.