Are These Functions Linearly Independent?

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Homework Help Overview

The discussion revolves around determining the linear independence of three functions: f1(x) = sqrt(x) + 5, f2(x) = sqrt(x) + 5x, and f3(x) = x - 1. Participants are exploring methods to prove this property.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • The original poster seeks assistance in proving linear independence and asks for guidance. Some participants suggest computing the Wronskian as a method to assess independence, while others emphasize the importance of recalling relevant definitions or theorems.

Discussion Status

The discussion is active, with participants sharing thoughts on potential methods for proving linear independence. There is a suggestion to use the Wronskian, which some consider a valid approach, but no consensus has been reached on the best method yet.

Contextual Notes

Participants are reminded to adhere to posting guidelines and to have considered relevant definitions or theorems before seeking help.

Naeem
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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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