Linear Independence/Dependence of set

In summary, the conversation discusses whether the set S* = span({x+z, x-y}) is linearly independent or dependent, given that S = {x,y,z} is linearly independent in a vector space V over a field F. The conversation includes a proof that the set is linearly dependent, using the example of choosing coefficients a and b to show that a linear combination of the two vectors in S* can equal zero. The conclusion is that the set S* is indeed linearly dependent.
  • #1
JG89
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Homework Statement



Suppose that x,y, and z are distinct vectors in a vector space V over a field F, and S = {x,y,z} is linearly independent. If S* = span({x+z, x-y}), prove whether S* is linearly independent or linearly dependent.

Homework Equations


The Attempt at a Solution



S* = a(x+z) + b(x-y) = ax + bx -by + az = (a+b)x - by + az, where a and b are coefficients. We know that this linear relationship only has the trivial solution (because we are told that the set S = {x,y,z} is linearly independent), thus S* must be linearly independent.This is the solution I have, but apparently it's wrong. The set is linearly dependent. Why is that?

EDIT: I know that I didn't write a linear combination of the span, I just wrote out the span. But the span itself is a linear combination of the vectors and a linear combination of a linear combination is still a linear combination, so just to simplify things I only wrote out the span of the vectors.
 
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  • #2
Is S* the whole span of those vectors, or is S* just the two vectors? If it's the span, it's obviously linearly dependent, as e.g. x+z, 2(x+z) are both in S*, so a(x+z) + b[2(x+z)] = 0 has b=-1, a=2 as a solution
 
  • #3
S* is the span. But the two vectors are (x+z) and (x-y). Not (x+z) and (x+z).
 
  • #4
Wow, I just realized what you were really trying to say. :rofl:

I see what you are saying now. But what is wrong with my proof?
 
  • #5
Your proof is that the two vectors x+z and x-y are linearly independent, which they are. The set of all vectors of the form a*(x+z) + b*(x-y) isn't... to check this, you'd take a look at a summation

[tex]\sum_{i=1}^n(a_i(x+z) + b_i(x-y)) = 0[/tex] for any scalars ai and bi (n is an arbitrary finite natural number)
 
  • #6
Thanks. I think I understand now (your first example made it easy)

Let me give my own example to see if I really have it down.

Let's say v = [tex]a_1(x+z) + b_1(x-y))[/tex] and

p = [tex]a_2(x+z) + b_2(x-y)) [/tex]

Then if you choose [tex]a_2 = -a_1[/tex] and [tex]b_2 = -b_1[/tex], then obviously the set consisting of v and p is linearly dependent.
 
  • #7
Sure, that's right. But you are still thinking too hard. I like (x+z) and 2(x+z) much better. Because it shows you how obvious it really is. The span contains an infinite numbers of vectors. A linearly independent set over a two dimensional subspace (like the span) contains only two. Even a subset containing three vectors MUST be linearly dependent. An infinite number is way overkill.
 

1. What is the definition of linear independence in a set?

Linear independence in a set refers to the property of a set of vectors where none of the vectors can be expressed as a linear combination of the other vectors in the set. In other words, no vector in the set can be written as a sum of the other vectors multiplied by some constants.

2. How do you test for linear independence of a set?

To test for linear independence of a set, you can use the determinant method or the rank method. The determinant method involves creating a matrix with the vectors in the set as columns and finding the determinant. If the determinant is non-zero, the set is linearly independent. The rank method involves creating a matrix with the vectors in the set as rows and finding the rank. If the rank of the matrix is equal to the number of vectors in the set, then the set is linearly independent.

3. Can a set with only two vectors be linearly dependent?

Yes, a set with only two vectors can be linearly dependent. This can happen if one of the vectors is a multiple of the other vector, or if the two vectors are parallel to each other.

4. What is the difference between linear independence and linear dependence?

The main difference between linear independence and linear dependence is that in a linearly independent set, none of the vectors can be expressed as a linear combination of the other vectors, while in a linearly dependent set, at least one vector can be written as a linear combination of the other vectors.

5. How does linear dependence/independence affect solutions to linear equations?

If a set of vectors is linearly independent, then there is only one solution to a system of linear equations involving those vectors. However, if the set is linearly dependent, there can be infinitely many solutions to the system, or no solutions at all.

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