Proving Linear Dependence in Spanned Vectors

In summary, the conversation discusses how to prove that if a vector v is in the span of a set of vectors v1, v2, ..., vN, then v and the other vectors in the set are linearly dependent. The attempt at a solution involves showing that there exists a set of scalars, not all equal to 0, that satisfy the homogeneous equation.
  • #1
XcKyle93
37
0

Homework Statement


This should be an easy one, I'm just making sure that I'm not screwing up horribly.

Prove that if v is in span(v1,v2, ..., vN), then v, v1, v2, ..., vN are linearly dependent.

Homework Equations


span(v1,v2, ..., vN) = {Ʃaivi}.

The Attempt at a Solution


If v is in span(v1,v2, ..., vN), then for some scalars a1, ..., aN, v = Ʃaivi. This means that 0 = Ʃaivi - v, which means that there exists a set of scalars, not all 0, that satisfy the homogeneous equation. This is because the scalar coefficient of v is -1.
 
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  • #2
XcKyle93 said:

Homework Statement


This should be an easy one, I'm just making sure that I'm not screwing up horribly.

Prove that if v is in span(v1,v2, ..., vN), then v, v1, v2, ..., vN are linearly independent.

Homework Equations


span(v1,v2, ..., vN) = {Ʃaivi}.

The Attempt at a Solution

.
If v is in span(v1,v2, ..., vN), then for some scalars a1, ..., aN, v = Ʃaivi. This means that 0 = Ʃaivi - v, which means that there exists a set of scalars, not all 0, that satisfy the homogeneous equation. This is because the scalar coefficient of v is -1.

You mean dependent of course. But your argument is OK.
 
  • #3
Okay, thanks. I was just making sure!
 

FAQ: Proving Linear Dependence in Spanned Vectors

What is basic linear algebra proof?

Basic linear algebra proof is a mathematical method used to prove theorems or propositions in linear algebra. It involves using logical deductions to show that a statement is true based on established definitions and axioms.

Why is basic linear algebra proof important?

Basic linear algebra proof is important because it allows us to understand the underlying principles and concepts of linear algebra, which is a fundamental branch of mathematics used in various fields such as physics, engineering, and computer science. It also helps us to develop critical thinking skills and problem-solving abilities.

How do I approach a basic linear algebra proof?

When approaching a basic linear algebra proof, it is important to carefully read and understand the statement you are trying to prove. Then, break it down into smaller, more manageable parts and use the definitions and theorems you have learned to make logical deductions and prove each part. Finally, combine all the smaller proofs to prove the original statement.

What are some common techniques used in basic linear algebra proof?

Some common techniques used in basic linear algebra proof include direct proof, proof by contradiction, proof by induction, and proof by cases. Additionally, techniques such as using algebraic manipulations, properties of matrices and vectors, and mathematical induction can also be helpful.

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To improve your skills in basic linear algebra proof, it is important to practice regularly and work on a variety of problems. You can also seek help from textbooks, online resources, and professors or tutors. It can also be beneficial to study and understand the proofs of well-known theorems in linear algebra to gain a deeper understanding of the subject.

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