My Math Forum Linear Independence Question #2

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 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
 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: non-pivot column =========== 5 2 3
December 23rd, 2017, 10:57 AM   #4
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Quote:
 Originally Posted by zollen . . . each row is not a combination of the others.
In your example, each column is a combination of the others.

 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.

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