Is a set with a 0 vector linearly independent?

In summary, a 0 vector is a vector with all components equal to 0, with no direction or magnitude. Its presence in a set is important for linear independence, as it serves as a reference point and ensures that no non-trivial linear combination of the other vectors in the set can equal 0. To determine if a set with a 0 vector is linearly independent, we must check if the only solution to the equation is when all coefficients are 0. A set with a 0 vector can never be linearly dependent, as its presence guarantees that there is no non-trivial linear combination resulting in 0. In linear algebra, a 0 vector is significant in understanding linear independence and is a starting point for other
  • #1
Sasor
16
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 hav any special properties with this?
 
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  • #2
If a set contains the zero vector, then it is always linearly dependent.

I think you made a typo when writing the title.
 

What does it mean for a set to have a 0 vector?

A 0 vector is a vector with all components equal to 0. In other words, it has no direction and no magnitude. It is often denoted as 0.

Why is the presence of a 0 vector important for linear independence?

A set with a 0 vector is important for linear independence because it serves as a reference point for other vectors in the set. If a set contains a 0 vector, it means that any linear combination of the other vectors in the set can never equal the 0 vector. This is a key criterion for linear independence.

How can I determine if a set with a 0 vector is linearly independent?

To determine if a set with a 0 vector is linearly independent, we can use the definition of linear independence. This means that we need to check whether the only solution to the equation a1v1 + a2v2 + ... + anvn = 0 is when all the coefficients ai are equal to 0.

Can a set with a 0 vector ever be linearly dependent?

No, a set with a 0 vector can never be linearly dependent. This is because the presence of a 0 vector guarantees that there is no non-trivial linear combination of the other vectors that can result in the 0 vector, which is a requirement for linear dependence.

What is the significance of a set with a 0 vector in linear algebra?

In linear algebra, a set with a 0 vector is important because it helps us understand the concept of linear independence. It also serves as a starting point for many other concepts, such as basis vectors and subspaces. Additionally, the presence or absence of a 0 vector in a set can greatly affect the properties and characteristics of that set.

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