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Help with figuring out Linear Dependency

  • #1

Homework Statement


Hello, I have been given a list of functions and I need to figure out if they are linearly dependent or not. I've been trying to solve the problems for a while but I cannot figure out which ones are wrong. The list of functions are on my webpage at: http://vantraveller.blogspot.ca/2013/09/linear-dependency-question.html [Broken]


Homework Equations



http://vantraveller.blogspot.ca/2013/09/linear-dependency-question.html [Broken]

The Attempt at a Solution


For sure, I know cos^2(x) and sin^2(x), x and x^2, 1+ln(x) and 1-ln(x) is linearly independent because they are not multiples of each other, but with regards to the others, I am not to sure.

Homework Statement





Homework Equations





The Attempt at a Solution

 
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Answers and Replies

  • #2
arildno
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Set up the equation that must govern linear (in)dependency.
Prove that the equation only holds when the coefficients for each function is set equal to zero.
 
  • #3
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For sure, I know cos^2(x) and sin^2(x), x and x^2, 1+ln(x) and 1-ln(x) is linearly independent because they are not multiples of each other, but with regards to the others, I am not to sure.
That's not a good reason. If you have two functions, it's easy to tell whether each is a multiple of the other, but with three or more functions, that thinking doesn't work any more. For example, {sin2(x), cos2(x), 1} is a linearly dependent set. No function is a multiple of any other in the set, but the equation c1 * sin2(x) + c2 * cos2(x) + c3 * 1 = 0 has a solution in which not all of the constants are zero.
 

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