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December 23rd, 2017, 07:58 AM   #1
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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.
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December 23rd, 2017, 10:21 AM   #2
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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".
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December 23rd, 2017, 10:32 AM   #3
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free variable = non pivot column

In the above example:

non-pivot column
===========
5
2
3
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December 23rd, 2017, 10:57 AM   #4
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Quote:
Originally Posted by zollen View Post
. . . each row is not a combination of the others.
In your example, each column is a combination of the others.
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December 23rd, 2017, 05:48 PM   #5
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I got it. The matrix A is linear dependent because it contains one free variable.
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December 23rd, 2017, 10:23 PM   #6
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Did the formal definitions that you were taught use the word "dependence" or "independence", or did they just use "dependent" or "independent"?
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December 23rd, 2017, 11:32 PM   #7
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linear dependent.
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December 24th, 2017, 12:43 AM   #8
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Okay. Some people use "linear dependent" and "linear independent", but I prefer "linearly dependent" and "linearly independent" on grammatical grounds.
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