Linear independence from functions

Functions can also be linearly independent

In addition to the vectors of two- or three-dimensional space that you are familiar with, there are other sets that meet the conditions of a vector space. An example are all continuous Functions over the real Counting R. (You do not necessarily need to know what the conditions for a vector space are to understand this further.)

• In a functional context, linear independence means that the set of functions f_i builds up or a complete subset of this. In other words: Any function, however arbitrary, can be used as a linear combination of these basic functions f_i represent.
• Just as you can test a set of vectors for linear independence, you can do the same with a set of functions. Put simply, a set of functions f_i then linearly independent if you cannot represent any of these functions as a linear combination of the other functions.
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• Mathematically, for linear independence it holds that the equation ∑ a_i * f_i = 0 can only be fulfilled if all (!) Real coefficients a_i = 0. This last mathematical expression is also a test criterion for the set of functions f_i. So in the end, just like with vectors, you have to find an equation with the unknowns a_i investigate.

Linear independence - examples

• An example often chosen for a set of continuous functions over R that are linearly independent is f₁(x) = x², f₂(x) = e^x and f₃(x) = e^-x. Even a preliminary consideration shows that none of these three functions can be expressed by the two remaining ones. Roughly speaking, the given functions are just too different. Also the equation a₁x² + a₂e^x * a₃e^-x = 0 can only be solved if all coefficients a_i = 0.


Linear combination of vectors - the math expert explains

You are confronted with the linear combination of vectors if you are in the ...

• The two functions f₁(x) = sin 2x, f₂However, (x) = sinx * cos x are linearly dependent, because you can convert the function of the double angle into the second function with the help of a formula.
• The (infinite) set of functions f_i(x) = xⁱ, where the index i is the numbers 0,1,2... runs through, by the way, forms a linearly independent basis of the vector space of the completely rational functions. The linear independence of f_i can be easily seen. The so-called Vronsky determinant.