Show that these vectors are in a vector space?

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Discussion Overview

The discussion centers on demonstrating that the set of vectors {u-v, v-w, w-u} is linearly dependent within a vector space V. Participants explore various methods and reasoning to establish this property in a general context.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to show that the vectors {u-v, v-w, w-u} are linearly dependent in a general sense.
  • Another participant suggests that vectors are dependent if a non-trivial linear combination equals zero and prompts for such a combination.
  • A participant proposes constructing a 3x3 matrix from the vectors and calculating its determinant to determine linear dependence or independence.
  • One participant attempts to express the vectors in terms of a new set {V1, V2, V3} and claims that one vector can be expressed as a sum of the others, suggesting linear dependence.
  • A participant provides a linear combination approach and sets up an augmented matrix to analyze the solution, concluding that the presence of a non-trivial solution indicates dependence.
  • Another participant reiterates the definition of dependent vectors and hints at a simple choice of coefficients that could demonstrate dependence.

Areas of Agreement / Disagreement

Participants generally agree on the goal of demonstrating linear dependence, but there are multiple approaches and methods proposed, with no consensus on a single method being definitive or correct.

Contextual Notes

Some participants express uncertainty about finding the determinant in a general form and the specifics of constructing the augmented matrix. The discussion includes various interpretations of linear dependence and the methods to demonstrate it.

ammar555
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How do I show that for any vectors u,v, and w in a vector space V, the set of the vectors {u-v, v-w, w-u} is a linearly dependent set?

do it in general!
 
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Isn't the point for you to "do it in general"? Vectors are "dependent" if and only if some linear combination of them (other than the trivial one with all multipliers equal to 0) is equal to 0. Can you find such a linear combination?
 
Just take the set of vectors, build a 3x3 matrix out of them and calculate its determinant. If the determinant is not equal to cero, then they are linearly independent. If it is cero, then they are linearly dependent.

EDIT: I have explained it better in this thread (actually the question is almost exactly the same)
 
Alpha Floor said:
Just take the set of vectors, build a 3x3 matrix out of them and calculate its determinant. If the determinant is not equal to cero, then they are linearly independent. If it is cero, then they are linearly dependent.

EDIT: I have explained it better in this thread (actually the question is almost exactly the same)


The teacher wants the answer in a general form, and I don't know how to find the determinant of that.

if I write something like this, is it correct?
{u-v, v-w, w-u} = {V1, V2, V3}

To prove this is linearly dependent, one of the vectors can be written as the sum of the other two vectors.

V3 = -V2-V1

Therefore, it is linearly dependent.
 
ammar555 said:
How do I show that for any vectors u,v, and w in a vector space V, the set of the vectors {u-v, v-w, w-u} is a linearly dependent set?

do it in general!

just add them up
 
Would this be correct?

a(u-v)+b(v-w)+c(w-u) = 0
au-av+bv-bw+cw-cu = 0

au-cu = 0
-au+bv = 0
-bw+cw = 0

augmented matrix =

1 0 -1 0
-1 1 0 0
0 -1 1 0

RREF on the calculator =

1 0 -1 0
0 1 -1 0
0 0 0 0

Since the soultion is not a trivial solution, it means the set is dependent.
 
The definition of 'dependent' vectors is that there exist a set if coefficients, not all 0, so that the linear combination is the 0 vector.

Is it possible to choose A, B, C, not all 0, so that A(u- v)+ B(v- w)+ C(w- u)= 0.
Hint- there is a very simple choice for A, B, and C.
 

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