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

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SUMMARY

The discussion clarifies that for linearly independent vectors y1 and y2, the ratio y2/y1 is not constant. This is because if y1 and y2 were linearly dependent, there would exist a scalar λ such that y1 = λy2. The context of the discussion suggests that y1 and y2 are functions belonging to a vector space related to solutions of a linear differential equation, emphasizing the importance of understanding the definitions of linear independence and dependence in this setting.

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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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