Show that these vectors are in a vector space?

In summary, to show that the set of vectors {u-v, v-w, w-u} is linearly dependent, we can use the definition that a set of vectors is dependent if there exists a non-trivial linear combination that equals 0. In this case, we can choose A=1, B=1, and C=1 to satisfy the linear combination and prove the set is dependent.
  • #1
ammar555
12
0
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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  • #2
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?
 
  • #3
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)
 
  • #4
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.
 
  • #5
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
 
  • #6
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.
 
  • #7
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.
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations that can be performed on those vectors. It is a fundamental concept in linear algebra and is used to model various physical and abstract systems.

2. How do you show that a set of vectors is in a vector space?

To show that a set of vectors is in a vector space, you must prove that the set satisfies the 10 axioms or properties of a vector space. These include closure under vector addition and scalar multiplication, existence of a zero vector, and existence of additive and multiplicative inverses for each vector.

3. What are the common examples of vector spaces?

Some common examples of vector spaces include the set of all real numbers, the set of all n-dimensional vectors, and the set of all polynomials of degree n or less. Other examples include function spaces, such as the set of all continuous functions or the set of all differentiable functions.

4. Can a vector space have infinitely many vectors?

Yes, a vector space can have infinitely many vectors. In fact, most vector spaces have an infinite number of vectors. For example, the set of all real numbers is an infinite vector space.

5. How are vector spaces used in science?

Vector spaces are used in various scientific fields, such as physics, engineering, and computer science. They provide a powerful mathematical framework for modeling and solving problems involving multidimensional data, physical forces, and transformations. Vector spaces are also essential in quantum mechanics, where they are used to describe the state of a quantum system.

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