Linear Independency: True/False Explained

[Precursor]

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.

 

[Dick]

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.

 

[Precursor]

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?

 

[Dick]

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.

 

[Precursor]

?? 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?

 

[Dick]

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?

 

[Precursor]

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.

 

[Dick]

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?

 

[Precursor]

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.

 

[Dick]

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.

 

[Precursor]

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?

 

[Dick]

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?

 

[Precursor]

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?

 

[Dick]

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.

 

[Mark44]

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.

 

[Precursor]

u = (0,2), v = (3,0), w = (0,1)

 

[Dick]

u = (0,2), v = (3,0), w = (0,1)

Bingo!