Linear algebra - Linear dependance.

[bobbyzilla]

Homework Statement

Suppose {u,v} and {v,w} are linearly dependent sets of vectors in R^n. Is {u,w} linearly dependent?

My Answer: don't know, individual vectors are neither dependent nor dependent, it depends on the context they are put in.

Solutions:
No, the sets {i,0} and {0,j} are each linearly dependent in R^2. however the set {i,j} is lienarly independent.

I have a problem rapping my head around this one. I visualize the vectors as column matricies but then the solution provided by the text just pisses me off. It doesn't explain anything, and i don't want to go visualize something like the xy yz and zx planes to determine if these veectors are dependent or not. is there another way to figure out the answer?

 

[Defennder]

Why doesn't it explain anything? For this question you either have to come up with a general proof for the claim or a counter-example to disprove it. It has clearly supplied a counter-example. Note that if any set of vectors contains the zero vector, then the set of vectors is considered a linearly dependent set, even if none of the non-zero vectors are linear combinations of each other. This follows from the standard definition of linear dependence of a set of vectors.

 

[bobbyzilla]

how do we know that the set {u,w} doesn't have a 0 in there to make the set linearly dependent ?

 

[Vid]

It can be either linearly independent or dependent. Two vectors u,v are linearly dep. iff u = c*v for some constant c. Similarly v,w are linearly dep. iff v = d*w for some constant d. For a nonzero constant c, v = 1/c*u = d*w. Thus, u = c*d*w and the two vectors are linearly dependent. However, as you've noticed if c = 0 then the two vectors u,w could be linearly independent or dependent.

 

[Defennder]

how do we know that the set {u,w} doesn't have a 0 in there to make the set linearly dependent ?

That's the thing. If none are non-zero vectors and if one isn't a multiple of the other, you've got 2 linearly independent vectors. But the moment you throw in a zero vector, the entire set becomes linearly dependent regardless of the other vectors already inside.

As for "how do we know", I suppose the question will hint to you in some way or in this case it asks you if a general conclusion can be drawn from knowing the set is linearly dependent. You have to consider the possbility of zero vector unless the question rules it out, which it doesn't do so here.

 

[bobbyzilla]

thanks dude,