Different ways to determine if functions are linearly independent

[find_the_fun]

Is calculating the determinant the Wronskian the only way to show that a set of functions is linearally independent? For example could you build a matrix where the numbers represent the coefficents to the polynomial functions and if it rref's to the identity matrix wouldn't this show it's linearally independent too?

Also, lots of the time can't you just use your brain? For example {2x, x} is dependent and {x^2+x, x} is independent. It seems calculating the Wronskian is a lot of extra work.

 

[gtoguy97]

No, calculating the determinant of the Wronskian is not the only way to show that a set of functions is linearly independent. It is possible to construct a matrix with coefficients that represent the polynomial functions and if it reduces to the identity matrix, then this could also show linear independence. However, it is generally easier to calculate the Wronskian to show linear independence. In some cases, it may be possible to determine linear independence simply by using one's intuition and examining the functions. However, this can be unreliable since linear dependence is not always obvious. Calculating the Wronskian is a more reliable method for determining linear independence.