Linear Independency: True/False Explained

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Homework Help Overview

The discussion revolves around the linear independence of vectors in R^n, specifically examining the claim that if two pairs of vectors are linearly independent, then a set of three vectors must also be linearly independent. Participants are tasked with determining the validity of this statement through examples and reasoning.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the conditions under which sets of vectors are linearly independent, questioning the implications of having two pairs of vectors that are independent. They discuss the necessity of providing counterexamples to validate or invalidate the original claim.

Discussion Status

The discussion is active, with participants offering examples and counterexamples to test the claim. Some guidance has been provided regarding the nature of linear independence and the role of the zero vector in forming independent sets. There is recognition of the need for further exploration to clarify the original statement.

Contextual Notes

Participants note the importance of the dimensionality of the vectors involved, with some suggesting that working in R^2 may simplify finding suitable examples. There is also mention of the generality of the statement across different dimensions.

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Homework Statement
Indicate whether the following is true or false. Explain your answer.

If \overline{u}, \overline{v}, \overline{w} are vectors in R^{n} such that {\overline{u}, \overline{v}} and {\overline{v}, \overline{w}} are each linearly independent sets, then {\overline{u}, \overline{v}, \overline{w}} is a linearly independent set.

The attempt at a solution
I think that the above is false because for {\overline{u}, \overline{v}} and {\overline{v}, \overline{w}} to each be linearly independent sets, they must have two entries for each vector, as this would give them trivial solutions only. Therefore, they would each be a 2 x 2 matrix. However, {\overline{u}, \overline{v}, \overline{w}} is not a linearly independent set because it would form a 3 x 2 matrix. This would automatically have a free variable, and so infinite solutions would result.
 
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Don't make this so complicated. Just give an example of three vectors where {u,v} and {v,w} are linearly independent, but {u,v,w} is not linearly independent. You can do it in R^2.
 
Dick said:
Don't make this so complicated. Just give an example of three vectors where {u,v} and {v,w} are linearly independent, but {u,v,w} is not linearly independent. You can do it in R^2.

Don't I have to do it in R^2?
 
Precursor said:
Don't I have to do it in R^2?

?? Do it in R^n where n is whatever. You just can't do it in R^1. Because {u,v} is always linearly dependent in R^1.
 
Dick said:
?? Do it in R^n where n is whatever. You just can't do it in R^1. Because {u,v} is always linearly dependent in R^1.

But isn't {u,v} a 2 x n matrix, where n must be two for the matrix to be linearly independent? If n was greater than 2, wouldn't you have more rows than columns, which means you would end up with free variables?
 
Precursor said:
But isn't {u,v} a 2 x n matrix, where n must be two for the matrix to be linearly independent? If n was greater than 2, wouldn't you have more rows than columns, which means you would end up with free variables?

That's where you are going wrong. {u,v} is not a matrix, it's just a list of two vectors. If u=(1,0,0), and v=(0,1,0) in R^3, is {u,v} linearly independent?
 
Dick said:
That's where you are going wrong. {u,v} is not a matrix, it's just a list of two vectors. If u=(1,0,0), and v=(0,1,0) in R^3, is {u,v} linearly independent?

{u,v} wouldn't be linearly independent since you end up with a row of all zeroes.
 
To show they are linearly dependent you want to find a solution to c1*u+c2*v=(0,0,0), where c1 and c2 are not both zero. Can you find one?
 
Dick said:
To show they are linearly dependent you want to find a solution to c1*u+c2*v=(0,0,0), where c1 and c2 are not both zero. Can you find one?

No. c1 = c2 = 0.
 
  • #10
Well, ok. So a 'row of zeros' has nothing to do with linear independence. Now back to the point. Can you find an example of three vectors where {u,v} and {v,w} are linearly independent and {u,v,w} is not.
 
  • #11
Dick said:
Well, ok. So a 'row of zeros' has nothing to do with linear independence. Now back to the point. Can you find an example of three vectors where {u,v} and {v,w} are linearly independent and {u,v,w} is not.

u = (1,0,1), v = (0,1,0), w = (0,0,1)

So this is a guess and check type of question? Is there a quicker way about this?
 
  • #12
Precursor said:
u = (1,0,1), v = (0,1,0), w = (0,0,1)

So this is a guess and check type of question? Is there a quicker way about this?

Mmm. You've got {u,v} and {v,w} independent AND {u,v,w} independent. Not really what you want. One more try, ok?
 
  • #13
Dick said:
Mmm. You've got {u,v} and {v,w} independent AND {u,v,w} independent. Not really what you want. One more try, ok?

Wait a second. This is a true or false question. Shouldn't the answer be "true" since {u,v,w} is independent?
 
  • #14
That's only one example. One example doesn't prove it's true. One counterexample will prove it's false. THINK ABOUT IT. You want {u,v} and {v,w} independent and {u,v,w} dependent. This isn't hard.
 
  • #15
Precursor said:
u = (3,2,1), v = (0,0,0), w = (1,2,3)
No linearly independent set can include the zero vector. So your example doesn't satisfy the hypothesis that {u, v} and {v, w} are lin. independent sets.

Try Dick's suggestion of working with vectors in R2. Then it should be easy to find three vectors where {u, v} and {v, w} are linearly independent sets, while {u, v, w} is a linearly dependent set. In fact, it will be difficult NOT to find three vectors for which this is true.

The statement you're working with is a sweeping generality for Rn, so if you can find a counterexample for a particular value of n -- say n = 2 -- then the entire statement is untrue.
 
  • #16
u = (0,2), v = (3,0), w = (0,1)
 
  • #17
Precursor said:
u = (0,2), v = (3,0), w = (0,1)

Bingo!
 

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