Linear algebra - Linear dependance.

In summary, the conversation discusses the concept of linear dependence of sets of vectors in R^n. It is noted that individual vectors can be either dependent or independent depending on the context. The solution provided in the text uses a counter-example to prove the claim that the set {u,w} can be linearly dependent even if the individual vectors {u,v} and {v,w} are not. The conversation also addresses the possibility of a zero vector in the set and how it affects the linear dependence.
  • #1
bobbyzilla
3
0

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 dependant nor independant, 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?
 
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  • #2
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.
 
  • #3
how do we know that the set {u,w} doesn't have a 0 in there to make the set linearly dependent ?
 
  • #4
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.
 
  • #5
bobbyzilla said:
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.
 
  • #6
thanks dude,
 

What is linear dependence?

Linear dependence is a concept in linear algebra that refers to a set of vectors that can be expressed as a linear combination of other vectors in the same vector space. In simpler terms, it means that one or more vectors in the set can be written as a combination of the other vectors in the set.

How do you determine if a set of vectors is linearly dependent or independent?

A set of vectors is linearly dependent if at least one of the vectors can be written as a linear combination of the others. To determine if a set of vectors is linearly dependent, you can use the following methods:

  • Row reduction: If the row-reduced echelon form of the matrix containing the vectors has a row of zeros, then the vectors are linearly dependent.
  • Determinant: If the determinant of the matrix containing the vectors is equal to 0, then the vectors are linearly dependent.
  • Linear combination: If you can find a non-trivial linear combination of the vectors that equals 0, then the vectors are linearly dependent.

Why is linear dependence important in linear algebra?

Linear dependence is important in linear algebra because it helps us understand the relationship between vectors in a vector space. It also allows us to determine if a set of vectors can be used to represent all other vectors in the vector space or if there are redundant vectors in the set.

What is the difference between linear dependence and linear independence?

Linear dependence and linear independence are two opposite concepts in linear algebra. As mentioned earlier, linear dependence refers to a set of vectors that can be written as a linear combination of other vectors in the same vector space. On the other hand, linear independence refers to a set of vectors that cannot be written as a linear combination of other vectors in the same vector space. In simpler terms, linear independence means that all the vectors in the set are unique and cannot be expressed as a combination of other vectors.

Can a set of two vectors be linearly dependent?

Yes, a set of two vectors can be linearly dependent. This means that one of the vectors can be written as a linear combination of the other vector. For example, if the two vectors are parallel, then one vector is a scalar multiple of the other, making them linearly dependent.

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