Linear Combinations of Dependent Vectors

In summary: If I was dealing with the vector space R^n, then x=αu+βv+γw would have infinitely-many solutions because you would end up with at least one free variable since at least one of the vectors is just a linear combination of the others. Can you apply that reasoning to all vector spaces?As you say, since (u,v,w) is linearly independent so there exist \alpha, \beta, \gamma, not all 0, such that \alpha u+ \beta v+ \gamma w= 0. Further, since (u,v,w) spans V there exist, for any x in V, scalars a
  • #1

Homework Statement



If (u,v,w) is a family of linearly dependent vectors in vector space V and vector x is in the span of (u,v,w), then x=αu+βv+γw has infinitely-many choices for α,β, and γ.

Homework Equations



If (u,v,w) is linearly dependent, then there exists an α, β, and γ, not all equal to zero, such that αu+βv+γw=0.

The Attempt at a Solution



My first attempt, which didn't go anywhere, was to assume that there were only a finite number of choices and see if that led to a contradiction.

For my second attempt, I started with the fact that (u,v,w) is linearly dependent. Then I multiplied both sides by an arbitrary scalar n. Then I thought I could add x to both sides and manipulate the equation somehow, but that didn't lead anywhere either.

If I was dealing with the vector space R^n, then x=αu+βv+γw would have infinitely-many solutions because you would end up with at least one free variable since at least one of the vectors is just a linear combination of the others. Can you apply that reasoning to all vector spaces?
 
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  • #2
As you say, since (u,v,w) is linearly independent so there exist [itex]\alpha[/itex], [itex]\beta[/itex], [itex]\gamma[/itex], not all 0, such that [itex]\alpha u+ \beta v+ \gamma w[/itex]= 0.

Further, since (u,v,w) spans V there exist, for any x in V, scalars a, b, c such that au+ bv+ cw= x.

Now, what can you say about [itex]au+ bv+ cw+ R(\alpha u+ \beta v+ \gamma w)[/itex] for any real number R?
 
  • #3
It equals x?

(Rα+a)u +(Rβ+b)v +(Rγ+c)w = x

And since Rα or Rβ or Rγ is not zero, the equation has infinitely-many solutions?
 
  • #4
Yes, every different choice for R gives a different linear combination but they are all equal to x!

And it is doing exactly what YOU suggested:
For my second attempt, I started with the fact that (u,v,w) is linearly dependent. Then I multiplied both sides by an arbitrary scalar n. Then I thought I could add x to both sides and manipulate the equation
 

What are linear combinations of dependent vectors?

Linear combinations of dependent vectors refer to the combination of two or more vectors that are dependent on each other. This means that they can be expressed as a multiple of each other, and one vector can be written as a linear combination of the others.

How do you determine if a set of vectors is linearly dependent or independent?

To determine if a set of vectors is linearly dependent or independent, you can use the concept of linear independence. A set of vectors is linearly independent if no vector in the set can be expressed as a linear combination of the other vectors. On the other hand, if one vector can be written as a linear combination of the others, the set is linearly dependent.

What is the significance of linear combinations of dependent vectors in linear algebra?

Linear combinations of dependent vectors are significant in linear algebra because they allow us to understand how a set of vectors is related to each other. They help us to determine if a set of vectors is linearly dependent or independent, and can be used to solve systems of linear equations.

Can a set of three or more vectors be linearly dependent?

Yes, a set of three or more vectors can be linearly dependent. In fact, any set of vectors with more than two vectors can be linearly dependent. This is because if one vector can be expressed as a linear combination of the others, the set is considered linearly dependent.

How can linear combinations of dependent vectors be used in real-world applications?

Linear combinations of dependent vectors can be used in many real-world applications, such as computer graphics, physics, and engineering. For example, in computer graphics, linear combinations of dependent vectors are used to create smooth and realistic animations. In physics, they are used to model the motion of objects, and in engineering, they are used to solve complex systems of equations.

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