# Help with figuring out Linear Dependency

• mikehsiao789
In summary, the conversation discusses determining if a list of functions is linearly dependent or not. The person has tried to solve the problems but is unsure about some of them. They mention that some functions are definitely linearly independent because they are not multiples of each other, but this reasoning does not work when there are more than two functions. The conversation also mentions setting up an equation to determine linear (in)dependency and proving that this equation only holds when the coefficients for each function are set equal to zero.

## 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.

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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.

mikehsiao789 said:
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.