# Linear Independence Question #2

#### zollen

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?

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#### johng40

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".

#### zollen

free variable = non pivot column

In the above example:

non-pivot column
===========
5
2
3

#### skipjack

Forum Staff
. . . each row is not a combination of the others.
In your example, each column is a combination of the others.

1 person

#### zollen

I got it. The matrix A is linear dependent because it contains one free variable.

#### skipjack

Forum Staff
Did the formal definitions that you were taught use the word "dependence" or "independence", or did they just use "dependent" or "independent"?

#### zollen

linear dependent.

#### skipjack

Forum Staff
Okay. Some people use "linear dependent" and "linear independent", but I prefer "linearly dependent" and "linearly independent" on grammatical grounds.