Is a set with a 0 vector linearly independent?

Click For Summary
SUMMARY

A set containing the zero vector is always linearly dependent. In the discussion, matrices such as [1 0 3], [2 0 4], and [0 0 5] are confirmed to span a plane in R³, demonstrating linear dependence. The matrices [1 0 3 5], [3 0 2 4], and [2 0 1 4] also exhibit linear dependence, spanning all of R³. The presence of the zero vector in any set guarantees that the set cannot be linearly independent.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically linear independence and dependence.
  • Familiarity with matrix representation and operations in R³.
  • Knowledge of vector spaces and their properties.
  • Ability to perform linear combinations of vectors.
NEXT STEPS
  • Study the properties of vector spaces in linear algebra.
  • Learn about the implications of the zero vector in linear combinations.
  • Explore examples of linearly independent and dependent sets in R³.
  • Investigate the concept of spanning sets and their significance in linear algebra.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching concepts of linear independence and dependence.

Sasor
Messages
15
Reaction score
0
I don't know how to write out matrices nicely on this forum,

but suppose you have some matrices:[1 0 3]
[2 0 4]
[0 0 5]

This would, by definition, be linearly dependent, spanning a plane in r3..is this correct? Since c1=0, c2=anything, c3=0

where c1v1+c2v2+c3v3=0

With this:

[1 0 3 5]
[3 0 2 4]
[2 0 1 4]

Linearly dependent, spanning all of r3?

[1 4 0 5 2]
[2 3 0 2 4]
[2 9 0 1 1]

Linearly dependent, spanning all of r3?

Are these correct? does the 0 vector have any special properties with this?
 
Last edited:
Physics news on Phys.org
If a set contains the zero vector, then it is always linearly dependent.

I think you made a typo when writing the title.
 

Similar threads

Undergrad Row space, Column space, Null space & Left null space
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Characterizing linear independence in terms of span
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Solve the system of linear equations
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Wronskian: null space equals the span of independent functions
  • · Replies 23 ·
Replies
23
Views
2K
MHB Statements with linearly independent vectors
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
  • · Replies 3 ·
Replies
3
Views
3K
MHB Are These Vectors Linearly Dependent?
  • · Replies 4 ·
Replies
4
Views
2K
MHB 12.6 linearly dependent or linearly independent?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Proving linear independence of two functions in a vector space
  • · Replies 5 ·
Replies
5
Views
2K