Why Is the Ratio of y2/y1 Not Constant for Linearly Independent Vectors?

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Discussion Overview

The discussion centers on the concept of linear independence in the context of vectors, specifically addressing why the ratio of two linearly independent vectors, y1 and y2, is not constant. The scope includes theoretical aspects of linear algebra and its implications in differential equations.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that if y1 and y2 are linearly independent, the ratio y2/y1 is not constant, but the reasoning behind this assertion is not provided.
  • Another participant suggests that if y1 and y2 are linearly dependent, there exists a scalar λ such that y1 = λy2, implying a relationship that would lead to a constant ratio.
  • A third participant inquires about the definitions of independence and dependence in the context of functions, indicating that y1 and y2 may belong to a vector space of functions related to a linear differential equation.
  • A later reply corrects a previous statement regarding the terminology, emphasizing that the discussion is about linear independence as a property of vectors within a vector space.

Areas of Agreement / Disagreement

Participants express differing views on the implications of linear independence and dependence, and the discussion remains unresolved regarding the specific reasoning behind the non-constant ratio of y2/y1.

Contextual Notes

The discussion may be limited by assumptions about the definitions of linear independence and dependence, as well as the specific context of the vector space being referenced.

Cantspel
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We were going over linear independents in class and my professor said that if y1 and y2 are linearly independent then the ratio of y2/y1 is not a constant, but he never explained why it is not a constant.
 
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Hi,
You can turn it around: if y1 and y2 are linearly dependent, there is a ##\lambda## such that ##y_1 = \lambda y_2##
 
Given that you posted this in a differential equations subforum, I take it that ##y_1## and ##y_2## belong to some vector space of functions that contains the solutions of a certain linear differential equation?

Provided this is indeed your setting, what (by definition) does it mean when ##y_{1,2}## are independent? What does it mean when they are dependent?
 
Cantspel said:
We were going over linear independents in class
Minor point -- you were going over linear independence in class. Linear independence is an attribute of a set of vectors of other elements that belong to a vector space.
 

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