
Linear Algebra Linear Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
December 23rd, 2017, 07:58 AM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 200 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 11:03 AM. 
December 23rd, 2017, 10:21 AM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
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, 10:32 AM  #3 
Senior Member Joined: Jan 2017 From: Toronto Posts: 200 Thanks: 2 
free variable = non pivot column In the above example: nonpivot column =========== 5 2 3 
December 23rd, 2017, 10:57 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 19,974 Thanks: 1850  
December 23rd, 2017, 05:48 PM  #5 
Senior Member Joined: Jan 2017 From: Toronto Posts: 200 Thanks: 2 
I got it. The matrix A is linear dependent because it contains one free variable.

December 23rd, 2017, 10:23 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 19,974 Thanks: 1850 
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, 11:32 PM  #7 
Senior Member Joined: Jan 2017 From: Toronto Posts: 200 Thanks: 2 
linear dependent.

December 24th, 2017, 12:43 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,974 Thanks: 1850 
Okay. Some people use "linear dependent" and "linear independent", but I prefer "linearly dependent" and "linearly independent" on grammatical grounds.


Tags 
independence, linear, question 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Linear Independence and Linear Dependence.  Luiz  Linear Algebra  7  August 29th, 2015 01:54 PM 
Linear Independence and Linear Dependence.  Luiz  Linear Algebra  1  August 26th, 2015 10:22 AM 
Linear Independence and Linear Dependence.  Luiz  Linear Algebra  5  August 24th, 2015 03:10 PM 
Linear Independence and Linear Dependence.  Luiz  Linear Algebra  1  August 24th, 2015 11:45 AM 
Linear independence & linear dependence  autumn94  Linear Algebra  1  March 29th, 2015 03:14 PM 