One set v is a linear combination of u. Prove u is linearly dependent

In summary: Yes. My Monday lecture did not spoke about bases or dimensions. But we did take about linear combination and linearly independent sets as well as a vector space.
  • #1
cbarker1
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Homework Statement
Let ##\{u_1,\ldots,u_s\}##and ##\{v_1,\ldots,v_t\}## be two sets of vectors. If ##s>t## and ##u_i## is a linear combination of ##v_1,\ldots, v_t##, show ##u_1,\ldots,u_s## is linearly dependent
Relevant Equations
Linear combination: c1u1+c2u2+c3u3+...+cnun=0
Linearly dependent is when there is a nontrival solution of the linear combination.
Hi Everybody,

I am having some difficulties on the prove this problem.
I picked a nice example when I was trying to think about the proof.

Let ##s=3## and ##t=2##. Then ##u1=c1v1+c2v2, u2=c3v1+c4v2, u3=c5v1+c6v2##. Then a linear combination of u: ##K1u1+K2u2+K3u3=0##. I grouped both linear combination of u in terms of v:
##K1(c1v1+c2v2)+K2(c3v1+c4v2)+K3(c5v1+c6v2)=0## It implies this system of linear equations for c1,..c6:
(K1c1+c3K2+c5K3)=0
(K1c2+c4K2+K3c6)=0
Here is where I am lost. What should I do next? Solve for K1,K2,K3? or something else.

Will this example help me prove this exercise?

Thanks,
cbarker 1
 
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  • #2
cbarker1 said:
Let ##s=3## and ##t=2##
This is strange. In the problem statement it says 'If ##s<t## ' ?
 
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  • #3
What if ##s=1 < 2=t## and ##u_1## is a straight in the plane spanned by ##\{v_1,v_2\}##. Then ##K\cdot u_1=0## implies ##K=0## and ##\{u_1\}## is linear independent.
 
  • #4
It seems quite obvious that the problem statement is wrongly reproduced and should say ##s>t## instead of ##s<t##.

cbarker1 said:
Homework Statement:: Let ##\{u_1,\ldots,u_s\}##and ##\{v_1,\ldots,v_t\}## be two sets of vectors. If ##s<t## and ##u_i## is a linear combination of ##v_1,\ldots, v_t##, show ##u_1,\ldots,u_s## is linearly dependent
Relevant Equations:: Linear combination: c1u1+c2u2+c3u3+...+cnun=0
Linearly dependent is when there is a nontrival solution of the linear combination.

Solve for K1,K2,K3? or something else.
You do not need to solve for the Ks. What you need to do is to argue that a non-trivial solution exists.

I might have gone down another route, but yours should work fine too once generalised.
 
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  • #5
I am an typo. I gave the wrong inequality symbol. Sorry
 
  • #6
Orodruin said:
It seems quite obvious that the problem statement is wrongly reproduced and should say ##s>t## instead of ##s<t##.You do not need to solve for the Ks. What you need to do is to argue that a non-trivial solution exists.

I might have gone down another route, but yours should work fine too once generalised.
What is the other route? I might easier to understand.
 
  • #7
I mean, it is in essence the same, but you get one constant less. Try to write ##u_s## as a linear combination of the first ##t## ##u_i##. (You can assume that the first ##t## ##u_i## are linearly independent because if they are not then you already know that the ##u_i## are linearly dependent.)
 
  • #8
Orodruin said:
I mean, it is in essence the same, but you get one constant less. Try to write ##u_s## as a linear combination of the first ##t## ##u_i##. (You can assume that the first ##t## ##u_i## are linearly independent because if they are not then you already know that the ##u_i## are linearly dependent.)

Proof by contradiction, right?
 
  • #9
cbarker1 said:
Proof by contradiction, right?
If you can use some basic results or theorems about bases and dimensions of linear spaces, then the proof follows without further explicit calculation.
 
  • #10
How can I do it without any theorems about bases and dimensions of linear spaces?
 
  • #11
cbarker1 said:
How can I do it without any theorems about bases and dimensions of linear spaces?
By explicit calculations that are similar to those used in the proofs of the relevant theorems.
 
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  • #12
cbarker1 said:
How can I do it without any theorems about bases and dimensions of linear spaces?
Is this problem presented in a vacuum? Aren't there any relevant theorems that can be used and that they expect you to use?
 
  • #13
Yes. My Monday lecture did not spoke about bases or dimensions. But we did take about linear combination and linearly independent sets as well as a vector space.
 
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1. What does it mean for one set v to be a linear combination of u?

A set v is said to be a linear combination of another set u if every element in v can be expressed as a linear combination of the elements in u. This means that each element in v can be written as a scalar multiple of an element in u, with all the scalar multiples added together.

2. How do you prove that u is linearly dependent?

To prove that u is linearly dependent, you need to show that there exists a non-zero solution to the equation c1u1 + c2u2 + ... + cnun = 0, where c1, c2, ..., cn are scalars and u1, u2, ..., un are the elements in u. This means that there is a non-trivial way to combine the elements in u to get a sum of zero.

3. What is the difference between linearly dependent and linearly independent sets?

A set is linearly dependent if there exists a non-trivial way to combine its elements to get a sum of zero. On the other hand, a set is linearly independent if there is no non-trivial way to combine its elements to get a sum of zero. In other words, the only way to get a sum of zero is by multiplying all the elements by zero.

4. Can you give an example of a linearly dependent set?

One example of a linearly dependent set is {v1, v2, v3} where v1 = (1, 2, 3), v2 = (2, 4, 6), and v3 = (3, 6, 9). In this set, v3 is a linear combination of v1 and v2, since v3 = 3v1 + 3v2. Therefore, this set is linearly dependent.

5. How is the linear dependence of a set related to its span?

The linear dependence of a set is closely related to its span. If a set is linearly dependent, it means that at least one of its elements can be written as a linear combination of the other elements. This means that the span of the set is not equal to the number of elements in the set. On the other hand, if a set is linearly independent, its span is equal to the number of elements in the set.

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