Are these sets of functions linearly dependent?

[mikehsiao789]

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

Homework Equations

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

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.

 

[arildno]

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.

 

[Mark44]

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.