Why Does Including the Zero Vector Make a Set Linearly Dependent?

In summary: For example, if b3 is added to the vector, then b3's coefficient is 0 plus the coefficient of b1, which is 1.
  • #1
dracolnyte
28
0

Homework Statement


Prove that any collection of vectors which includes [tex]\theta[/tex] (zero vector or null vector)is linearly dependent. Thus, null vector cannot be contained in a basis.


The Attempt at a Solution


Well, I know that in order for a collection of vectors to to be linearly dependent, one vector can be expressed as a linear combination of other vectors such as:
let s be some non-zero scalar
let v be vectors

s1v1 + s2v2 + ... + skvk = 0

but let's say that v2 was a zero vector (is this what the question is asking?),
-s2v2 = s1v1 + s3v3 + ... + skvk?
I don't quite get the phrase "any collection of vectors which includes 0(theta)"
 
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  • #2
Start off by proving the set containing ONLY the zero vector is linearly dependent. It's easiest to use the direct definition here: A set of vectors v1,..., vn is linearly dependent if and only if there exist coefficients s1,..., sn such that not all the si are zero (but some of them are allowed to be) and that

s1v1 + ... + snvn = 0

If you only have the zero vector, what s1 can you pick to satisfy this? Then consider if you add additional vectors... what coefficients can you pick for them?
 
  • #3
Okay thanks, I got that one. How about this one?

1. Homework Statement
Prove that if the vectors b1, b2,...bm are linearly dependent, then any collection of vectors which contains the b's must also be linearly dependent

3. The Attempt at a Solution
So to be linearly dependent,
s1b1 + s2b2 +... + smbm = 0
given that the scalar s is not equal to zero.

How should I go about proving this one?
 
  • #4
dracolnyte said:
Okay thanks, I got that one. How about this one?

1. Homework Statement
Prove that if the vectors b1, b2,...bm are linearly dependent, then any collection of vectors which contains the b's must also be linearly dependent

3. The Attempt at a Solution
So to be linearly dependent,
s1b1 + s2b2 +... + smbm = 0
given that the scalar s is not equal to zero.
There is NO "scalar s" in what you wrote! You mean that "at least one of the s1, s2, ..., sm is not 0."

How should I go about proving this one?
If a collection of vectors includes the b's, take the coefficients of the addtional vector to all be 0.
 

Related to Why Does Including the Zero Vector Make a Set Linearly Dependent?

1. What is the purpose of constructing a basis in science?

A basis is a fundamental set of elements or principles that serve as a starting point for further scientific research and understanding. It helps to establish a framework for analyzing and interpreting data, making predictions, and testing hypotheses.

2. How is a basis constructed in science?

A basis is constructed by identifying and selecting a set of fundamental principles, theories, or concepts that are widely accepted and have been extensively tested and supported by evidence. These elements form the foundation upon which further scientific knowledge is built.

3. Why is it important to have a well-defined basis in science?

A well-defined basis provides a solid and reliable foundation for scientific inquiry, ensuring that research and findings are based on sound principles. It also allows for easier communication and collaboration among scientists, as they can refer back to a common set of principles and assumptions.

4. Can a basis change or evolve over time?

Yes, a basis in science can change or evolve over time as new evidence is discovered and theories are refined. However, any changes to the basis should be well-supported and thoroughly tested to ensure its validity and reliability.

5. How does the construction of a basis impact the overall scientific process?

The construction of a basis is a crucial step in the scientific process as it provides a solid foundation for further research and discovery. A well-constructed basis helps scientists to ask meaningful questions, design experiments, and interpret results accurately, leading to a deeper understanding of the natural world.

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