Are These Functions Linearly Independent?

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To determine if the functions f1(x) = sqrt(x) + 5, f2(x) = sqrt(x) + 5x, and f3(x) = x - 1 are linearly independent, one suggested method is to compute the Wronskian. If the Wronskian is equal to zero, the functions are linearly dependent; if it is non-zero, they are independent. The discussion emphasizes the importance of understanding the definition of linear independence and relevant theorems. Participants are encouraged to engage with the problem and apply mathematical principles to reach a conclusion. The focus remains on using the Wronskian as a reliable tool for this proof.
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Q. Prove whether or not the following are linearly independent.

1. f1 (x) = sqrt (x) + 5

2. f2(x) = sqrt (x) + 5x

3. f3(x) = x-1

How do we prove these?

Can anyone help
 
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https://www.physicsforums.com/showthread.php?t=4825

You must have had a thought on this problem already -- surely you know, say, the definition, or a relevant theorem?
 
My thought is to compute the wronskian:

and see if it is equal = 0 it is dependent, if not independent.
 
That would be an excellent way to do it--- you would do well to recall the 3x3 determinant.
 

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