Linear Independence of subsets

In summary, the problem is to show that any subset of a linearly independent set of vectors is also linearly independent. This means that if the vector equation x1v1+...+xpvp=0 has only the trivial solution, then any subset of these vectors will also have only the trivial solution for the same equation. This can be proven by contradiction, assuming that a nontrivial solution exists for the subset and showing that this leads to a nontrivial solution for the original set of vectors. Additionally, it is important to note that the definition of linear independence only allows for one or no solutions, and the zero vector solution is known as the trivial solution.
  • #1
topgear
10
0

Homework Statement


Suppose {V1, V2, ..., Vp} form a linearly independent set of vectors. Show that any subset of this collection of vectors is also linearly independent. Is it necessarily true that is the vectors are dependent, that any subset is also dependent?


Homework Equations


The 10 axioms of subspaces
5 facts about subspaces


The Attempt at a Solution


Is a subset and a subspace the same thing? It makes since that if you have something in something that has finite answers the subset wouldn't have infinite answers. I am just having trouble proving it.
 
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  • #2
topgear said:
Is a subset and a subspace the same thing?

No, not at all! {2} is a subset of [tex]\mathbb{R}[/tex], but it isn't a subspace!

It makes since that if you have something in something that has finite answers the subset wouldn't have infinite answers. I am just having trouble proving it.

I'm having trouble seeing what you're getting at...

Like any problem, we need to start with the definitions. What does "linear independent" mean? How is it defines? Do you know equivalent properties for it?
 
  • #3
Linear Independent: means it has only one or no answers
Would a subset be {v1,v2,v3}, only part of {v1,v2,...,vp}?
 
  • #4
topgear said:
Linear Independent: means it has only one or no answers

Sorry, this makes no sense to me. Could you please copy the exact definition?
 
  • #5
An indexed set of vectors {v1,v2,...vp} in R^n is said to be linearly independent if the vector equation x1v1+x2v2+...+xpvp=0 has only the trivial solution.

The zero vector solution is known as the trivial solution.
 
  • #6
Indeed, so you know that [tex]x_1v_1+...+x_pv_p=0[/tex] only has the trivial solution. Now, we we take [tex]\{v_{i_1},...,v_{i_n}\}\subseteq \{v_1,...,v_p\}[/tex]. And we need to show that [tex]x_{i_1}v_{i_1}+...+x_{i_p}v_{i_p}=0[/tex] only has a trivial solution. Maybe you should try contradiction here? Assume that the latter equation has a nontrivial solution, then make a solution for the first equation...
 
  • #7
I'm sorry if I'm being slow I'm a graphic design major trying to take this class because my parents don't think art is a good major and I need math a math minor and it's killing me.

You're saying I should prove it by If P then not Q. So If x1v1+...xpvp=1 then xi1vi1+...+xipvip is non trivial? Doesn't that make it If not P then not Q?
 
  • #8
No, you should prove "If not Q, then not P"...

I actually wonder why people consider math a good degree and graphic design a bad degree. That makes no sense to me. Graphic design is much more applied and real-worldy than math. Strange...
 

FAQ: Linear Independence of subsets

What is linear independence of subsets?

Linear independence of subsets refers to a property of a set of vectors in a vector space, where no vector in the set can be represented as a linear combination of the other vectors. In other words, the vectors in a linearly independent subset cannot be "duplicated" or "created" by any combination of the other vectors in the set.

Why is linear independence important?

Linear independence is important because it allows us to determine the dimension of a vector space. If a set of vectors is linearly independent, then the number of vectors in the set is equal to the dimension of the vector space. Additionally, linear independence is a necessary condition for a set of vectors to form a basis for a vector space.

How can we test for linear independence?

To test for linear independence, we can use the definition of linear independence: a set of vectors is linearly independent if and only if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set, is the trivial solution where c1 = c2 = ... = cn = 0. We can also use Gaussian elimination or row reduction to determine if the vectors are linearly independent.

What is the difference between linear independence and linear dependence?

Linear independence and linear dependence are two opposite concepts. As mentioned earlier, linear independence refers to a set of vectors where none of the vectors can be represented as a linear combination of the other vectors in the set. Linear dependence, on the other hand, refers to a set of vectors where at least one vector can be represented as a linear combination of the other vectors in the set.

Can a set of two vectors be linearly independent?

Yes, a set of two vectors can be linearly independent. In fact, any set of two non-zero vectors in a two-dimensional vector space is linearly independent. However, in a higher-dimensional vector space, a set of two vectors may or may not be linearly independent, depending on the specific vectors in the set.

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