Different ways to determine if functions are linearly independent

In summary, calculating the Wronskian is not the only way to show linear independence, but it is generally the most efficient and reliable method.
  • #1
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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.
 
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  • #2
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.
 

Related to Different ways to determine if functions are linearly independent

1. How do you determine if functions are linearly independent?

To determine if functions are linearly independent, you can use the Wronskian determinant or the method of reduction of order. The Wronskian determinant involves calculating the determinant of a matrix composed of the functions and their derivatives, while the method of reduction of order involves setting up a system of equations and solving for the coefficients to see if they are all equal to 0.

2. Can you explain the Wronskian determinant method?

The Wronskian determinant method involves constructing a matrix with the functions and their derivatives as its rows or columns. If the determinant of this matrix is equal to 0, then the functions are linearly dependent. If the determinant is not equal to 0, then the functions are linearly independent.

3. What is the purpose of the method of reduction of order?

The method of reduction of order is used to determine if a set of functions are linearly independent by setting up a system of equations and solving for the coefficients. If the coefficients are all equal to 0, then the functions are linearly dependent. If the coefficients are not all equal to 0, then the functions are linearly independent.

4. Is there a difference between linear independence and linear dependence?

Yes, there is a difference between linear independence and linear dependence. Linear independence refers to a set of functions that cannot be expressed as a linear combination of each other. On the other hand, linear dependence refers to a set of functions that can be expressed as a linear combination of each other.

5. Can you give an example of linearly independent functions?

Yes, a set of linearly independent functions could be f(x) = x, g(x) = x^2, and h(x) = x^3. These functions cannot be expressed as a linear combination of each other. However, a set of linearly dependent functions could be f(x) = x, g(x) = 2x, and h(x) = 3x. In this case, h(x) is equal to 3f(x) + 3g(x), making the functions linearly dependent.

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