Linear Independence: Matrix/Equations Analysis

Click For Summary

Homework Help Overview

The discussion revolves around the concept of linear independence and dependence in the context of matrices and systems of equations. Participants are examining a specific matrix and questioning how to determine its linear independence or dependence.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are exploring definitions of linear independence and dependence, questioning the relationship between the number of components in vectors and the independence of rows and columns in the matrix. There is also a discussion about the method of row reduction as a means to check for dependence.

Discussion Status

The discussion is ongoing, with participants seeking clarification on terminology and concepts. Some guidance has been offered regarding the relationship between the number of vectors and their components, as well as the method of row reduction to assess dependence.

Contextual Notes

There is a mention of the matrix having more columns than rows, which raises questions about the linear dependence of the columns. Participants are also clarifying the meaning of "components" in relation to the matrix's structure.

TheColorCute
Messages
22
Reaction score
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! :)
 
Physics news on Phys.org
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.
 
What do you mean by "3 components"?
 
TheColorCute said:
What do you mean by "3 components"?

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

Ahhh. And by "components" you mean rows?
 

Similar threads

  • · Replies 30 ·
2
Replies
30
Views
2K
Replies
2
Views
1K
Replies
1
Views
2K
  • · Replies 3 ·
Replies
3
Views
1K
  • · Replies 2 ·
Replies
2
Views
2K
Replies
4
Views
2K
Replies
4
Views
2K
  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K