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December 23rd, 2017, 06:58 AM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 169 Thanks: 2  Linear Independence Question #2
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 skipjack; December 23rd, 2017 at 10:03 AM. 
December 23rd, 2017, 09:21 AM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 82 Thanks: 40 
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". 
December 23rd, 2017, 09:32 AM  #3 
Senior Member Joined: Jan 2017 From: Toronto Posts: 169 Thanks: 2 
free variable = non pivot column In the above example: nonpivot column =========== 5 2 3 
December 23rd, 2017, 09:57 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,704 Thanks: 1529  
December 23rd, 2017, 04:48 PM  #5 
Senior Member Joined: Jan 2017 From: Toronto Posts: 169 Thanks: 2 
I got it. The matrix A is linear dependent because it contains one free variable.

December 23rd, 2017, 09:23 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,704 Thanks: 1529 
Did the formal definitions that you were taught use the word "dependence" or "independence", or did they just use "dependent" or "independent"?

December 23rd, 2017, 10:32 PM  #7 
Senior Member Joined: Jan 2017 From: Toronto Posts: 169 Thanks: 2 
linear dependent.

December 23rd, 2017, 11:43 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 18,704 Thanks: 1529 
Okay. Some people use "linear dependent" and "linear independent", but I prefer "linearly dependent" and "linearly independent" on grammatical grounds.


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