Linear Independence: Matrix/Equations Analysis

[TheColorCute]

How do you know when a matrix (or equivocally a system of equations) is linearly independent? How do you know that it's linearly dependent?

For example, given this matrix,

[ 1 1 2 1]
[-2 1 4 0]
[ 0 3 2 2]

How do we know if this matrix is linearly independent or dependent?

Thanks! :)

 

[LCKurtz]

How do you know when a matrix (or equivocally a system of equations) is linearly independent? How do you know that it's linearly dependent?

For example, given this matrix,

[ 1 1 2 1]
[-2 1 4 0]
[ 0 3 2 2]

How do we know if this matrix is linearly independent or dependent?

Thanks! :)

It isn't a matrix that is linearly dependent or independent. You can ask whether its rows or columns are. In this case the columns must be dependent because there are 4 of them and the columns have 3 components. To check whether the rows are dependent you would do row reduction. If a row becomes all 0 the rows are dependent.

 

[TheColorCute]

What do you mean by "3 components"?

 

[LCKurtz]

What do you mean by "3 components"?

Each column is a 3d column vector. It has, count 'em, three components.

 

[TheColorCute]

Each column is a 3d column vector. It has, count 'em, three components.

Ahhh. And by "components" you mean rows?