Please help to prove this theorem

  • Thread starter loli12
  • Start date
  • Tags
    Theorem
In summary, the conversation discusses the use of summation notation in proving a theorem related to linear independence. The speaker doubts that avoiding the use of the summation sign is the best approach and provides an example using summation notation to prove the theorem. The speaker recommends trying to understand and use summation notation to gain a better understanding of the concept.
  • #1
loli12
Can someone help me prove this theorem without using the summation sign? because the proof in my book uses the summation method to prove this and i have trouble understanding the those signs. Thanks a lot!

Theorem: If {V1, V2,...Vn} is a spanning set for a vector space V, then any colletion of m vectors in V, where m>n, is linearly dependent.
 
Physics news on Phys.org
  • #2
Actually, I doubt it. Saying that you do not understand a particular symbol doesn't mean using it isn't the best way to do a problem.

Certainly, you can give examples that might be clearer but don't fool yourself into thinking that that is the same as a proof!

Suppose {V1, V2, V3} is a spanning set for a vector space V and {u1, u2, u3, u4} is a collection of vectors in V. (Here, I'm taking n= 3, m= 4> 3.)

Now, suppose you have some numbers, a, b, c, d, so that au1+ bu2+ cu3+ du4= 0.

Since V1, V2, V3 span V, we can write each of u1, u2, u3, u4 using those:
u1= p1V1+ p2V2+ p3V3 for some numbers p1, p2, p3
u2= q1V1+ q3V2+ q3V3 for some numbers q1, q2, q3
u3= r1V1+ r2V2+ r3V3 for some numbers r1, r2, r3
u4= s1V1+ s2V2+ s3V3 for some numbers s1, s2, s3

Now put those into the equation au1+ bu2+ cu3+ du4= 0:
a(p1V1+ p2V2+ p3V3)+ b(q1V1+ q2V2+ q3V3)+c(r1V1+ r2V2+ r3V3+d(s1V1+ s2V2+ s3V3)= 0.

we can write that as
(ap1+ bq1+ cr1+ ds1)V1+ (ap2+ bq2+cr2+ds2)V2+ (ap3+bq3+cr3+ds3)V3= 0

Certainly one way that can be true (not necessarily the only way- we are not assuming V1, V2, V3 are independent themselves) is if each coefficient is 0:
ap1+ bq1+ cr1+ ds1= 0
ap2+ bq2+ cr2+ ds2= 0
ap3+ bq3+ cr3+ ds3= 0

That's three homogenous equations for 4 unknowns, a, b, c, d. Certainly a= b= c= d= 0 is one solution but its easy to see that there are others. For example, take d= 0, c= 1 Then we have ap2+ bq2= -r2, ap3+ bq3= -r3 and can solve those two equations for non-zero a and b.

That's an example to illustrate how the general proof works. If you want to prove it is true for any n, you HAVE to use some kind of general notation. I recommend the you rewrite this example using summation notation to get a better idea of how summation notation works!
 
  • #3
Thanks a lot. I will try using the summation notation with it!
 

1. What is the importance of proving a theorem?

Proving a theorem is important because it allows us to establish the truth or validity of a mathematical statement. It also helps us to understand the underlying principles and concepts behind the statement, and how it relates to other mathematical ideas.

2. How do you approach proving a theorem?

The approach to proving a theorem involves breaking down the statement into smaller, more manageable pieces and then using logical reasoning and mathematical techniques to connect these pieces together to form a coherent proof. It also requires creativity and critical thinking to come up with new ideas and ways to approach the problem.

3. What makes a good proof for a theorem?

A good proof for a theorem is one that is clear, concise, and logically sound. It should also be easily understandable by others and should provide insight into the underlying principles and concepts of the statement being proved. Additionally, a good proof should be able to withstand scrutiny and be applicable to different scenarios.

4. How long does it take to prove a theorem?

The time it takes to prove a theorem can vary greatly depending on the complexity of the statement, the approach used, and the mathematician's expertise in the subject. Some theorems can be proved relatively quickly, while others may take years or even decades to solve.

5. Can a theorem ever be proven wrong?

Yes, a theorem can be proven wrong if it is based on incorrect assumptions or if a mistake is made in the proof. In mathematics, there is always the possibility of discovering new information or techniques that can challenge previously proven theorems. Therefore, it is important for mathematicians to constantly review and refine their proofs to ensure their validity.

Similar threads

  • Linear and Abstract Algebra
Replies
2
Views
493
  • Linear and Abstract Algebra
Replies
3
Views
963
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
10
Views
2K
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
10
Views
3K
  • Linear and Abstract Algebra
Replies
1
Views
997
  • Linear and Abstract Algebra
Replies
11
Views
2K
  • Linear and Abstract Algebra
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
14
Views
1K
Back
Top