Linear Independence: Matrix/Equations Analysis

Yes, the components of a column vector would be the rows of the matrix.In summary, to determine if a matrix or system of equations is linearly independent or dependent, you can check if its rows or columns are dependent. This can be done by row reduction or checking for all 0 rows. The number of components (or rows) in each column vector can also indicate dependence.
  • #1
TheColorCute
22
0
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! :)
 
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  • #2
TheColorCute said:
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.
 
  • #3
What do you mean by "3 components"?
 
  • #4
TheColorCute said:
What do you mean by "3 components"?

Each column is a 3d column vector. It has, count 'em, three components.
 
  • #5
LCKurtz said:
Each column is a 3d column vector. It has, count 'em, three components.

Ahhh. And by "components" you mean rows?
 

FAQ: Linear Independence: Matrix/Equations Analysis

What is linear independence?

Linear independence refers to the relationship between vectors in a vector space. A set of vectors is considered linearly independent if none of the vectors can be expressed as a linear combination of the other vectors in the set. In other words, no vector in the set can be written as a combination of the other vectors in the set using scalar multiplication and addition.

How is linear independence related to matrices?

In linear algebra, matrices can represent systems of linear equations. A set of equations is considered linearly independent if none of the equations can be derived from the others. This is equivalent to the concept of linear independence in vector spaces. Therefore, linear independence is an important concept in matrix analysis, as it helps determine the solvability of a system of equations.

How do you test for linear independence?

To test for linear independence, we can use the determinant of a matrix. If the determinant is non-zero, then the set of vectors (or equations) is linearly independent. In other words, the system of equations has a unique solution. However, if the determinant is zero, then the set of vectors is linearly dependent, and the system of equations has either no solution or infinitely many solutions.

What is the significance of linear independence in data analysis?

In data analysis, linear independence is crucial for understanding the relationships between variables. If two variables are linearly dependent, it means that one variable can be predicted from the other, and therefore, there is no new information gained by including both variables in a model. On the other hand, if two variables are linearly independent, they both provide unique information and can improve the accuracy of a model.

Can a set of vectors be both linearly independent and linearly dependent?

No, a set of vectors cannot be both linearly independent and linearly dependent. If a set of vectors is linearly dependent, it means that at least one vector can be expressed as a linear combination of the others. In contrast, linearly independent vectors cannot be written as a linear combination of each other. Therefore, these two concepts are mutually exclusive.

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