Is it possible to prove linear dependence using a simple approach?

In summary, the conversation discusses the proof that if {v_1, v_2, v_3} is a linearly dependent set of vectors in \mathbb{R}^n, then {v_1, v_2, v_3, v_4} is also linearly dependent, where v_4 is any other vector in \mathbb{R}^n. The conversation includes hints and explanations on how to approach the proof, as well as clarifications on the definition of linear dependence. Ultimately, it is concluded that the second set will also be linearly dependent because the weights c_1, c_2, ... c_n cannot all be equal to zero at the same time.
  • #1
tandoorichicken
245
0
This may be a really simple proof but its giving me grief.

If {[itex]v_1, v_2, v_3[/itex]} is a linearly dependent set of vectors in [itex]\mathbb{R}^n[/itex], show that {[itex]v_1, v_2, v_3, v_4[/itex]} is also linearly dependent, where [itex]v_4[/itex] is any other vector in [itex]\mathbb{R}^n[/itex].

Any hints on where to start? I started out by writing out all the claims that could be made by taking the first set of vectors to be linearly independent, but that didn't get me terribly far. :confused:
 
Physics news on Phys.org
  • #2
The definition of lineair dependence (or at least an equivalent statement) is that you can form the zero-vector with a lineair combination of your vectors without having all 0 coefficients. Now, you can do that for your first set, would you be able to do it too for the second one?
 
  • #3
i don't like this question because it doesn't exlude a v_4 that is in the span of the first set of vectors.

in that case, the second set WOULD be dependent!

 
  • #4
Brad Barker said:
in that case, the second set WOULD be dependent!

But that's exactly what you want to prove, so your proof is finished!
 
  • #5
I think I got it. The rule basically says the weights c_1, c_2, ... c_n can't all be equal to zero at the same time. If the first set is already linearly dependent, i.e. [itex]c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 = 0[/itex], then we can take c_4 = 0, which would make [itex]c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 + c_4\vec{v}_4 = 0[/itex] as well, therefore the second set is also linearly dependent.
 
  • #6
That's what I meant :smile:
 
  • #7
oh!

i thought it said INdependent!

:smile:
 
  • #8
tandoorichicken said:
I think I got it. The rule basically says the weights c_1, c_2, ... c_n can't all be equal to zero at the same time. If the first set is already linearly dependent, i.e. [itex]c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 = 0[/itex], then we can take c_4 = 0, which would make [itex]c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 + c_4\vec{v}_4 = 0[/itex] as well, therefore the second set is also linearly dependent.

yep, that's it. good job.
 

FAQ: Is it possible to prove linear dependence using a simple approach?

1. What is linear dependence?

Linear dependence is a mathematical concept that describes the relationship between a set of vectors. It means that one or more of the vectors can be written as a linear combination of the others, meaning they can be expressed as a constant multiple of each other. This indicates that the vectors are not independent and do not add any new information to the set.

2. How is linear dependence determined?

Linear dependence can be determined by performing row reduction on a matrix composed of the vectors in question. If a row of zeros is obtained during the row reduction process, it indicates that the vectors are linearly dependent. Another method is to calculate the determinant of the matrix, and if it is equal to zero, the vectors are linearly dependent.

3. What is the difference between linear dependence and linear independence?

Linear independence is the opposite of linear dependence. It means that the vectors in a set are not related to each other by a linear combination and are therefore independent. This indicates that each vector in the set provides unique information and cannot be expressed as a multiple of any other vector.

4. Why is proving linear dependence important in mathematics?

Proving linear dependence is important because it helps us understand the relationships between vectors and determine if they are providing new information or redundant information. It is also a fundamental concept in linear algebra and is used in various mathematical applications such as solving systems of equations, calculating eigenvalues and eigenvectors, and finding the basis of a vector space.

5. Can a set of only two vectors be linearly dependent?

Yes, a set of only two vectors can be linearly dependent. This is because if the two vectors are multiples of each other, they are considered linearly dependent. For example, if one vector is twice the other, it can be expressed as 2 times the first vector, making them linearly dependent.

Similar threads

Linear (in)dependence of vector pairs and triples
Replies
7
Views
2K
Can Direct Sums and Projections Fully Describe Subspaces in Linear Algebra?
Replies
5
Views
1K
Isomorphisms preserve linear independence
Replies
2
Views
2K
Dimension of orthogonal subspaces sum
Replies
4
Views
1K
Given specific v, dimension of subspace of L(V, W) where Tv=0?
Replies
15
Views
2K
Linear Combination Proof of Orthonormal basis
Replies
11
Views
2K
A quite verbal proof that if V is finite dimensional then S is also....
Replies
34
Views
2K
Null space of dual map of T = annihilator of range T
Replies
1
Views
974
Dimensionality of the sum of subspaces
2
Replies
45
Views
3K
Do these two statements imply an underlying induction proof?
Replies
7
Views
792

Hot Threads

Recent Insights

Back
Top