Proving Linear Dependence in Spanned Vectors

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SUMMARY

The discussion centers on proving that if a vector v is in the span of vectors v1, v2, ..., vN, then the set {v, v1, v2, ..., vN} is linearly dependent. The proof utilizes the definition of span, represented as span(v1, v2, ..., vN) = {Ʃaivi}, and demonstrates that the equation 0 = Ʃaivi - v indicates the existence of non-zero scalars, confirming linear dependence. The participants clarify that the original statement mistakenly referred to linear independence instead of dependence, affirming the correctness of the argument presented.

PREREQUISITES
  • Understanding of vector spaces and linear combinations
  • Familiarity with the concept of span in linear algebra
  • Knowledge of linear dependence and independence
  • Ability to manipulate and solve homogeneous equations
NEXT STEPS
  • Study the properties of vector spaces in linear algebra
  • Learn about the implications of linear independence and dependence
  • Explore examples of spans and their geometric interpretations
  • Practice solving homogeneous equations in various contexts
USEFUL FOR

Students studying linear algebra, educators teaching vector spaces, and anyone interested in understanding the foundational concepts of linear dependence and span in mathematics.

XcKyle93
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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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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.
 
Okay, thanks. I was just making sure!
 

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