# Linear Combination Function

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A Linear Combination Function is a mixture function constructed from a set of terms by multiplying each term by a constant and adding the results.

**AKA:**Linear Mixture.**Example(s):****Counter-Example(s):****See:**Mixture Model, Principal Component Analysis, Linear Classifier, Linear Operation, Linear Function, Linear Classification Model.

## References

### 2011

- http://en.wikipedia.org/wiki/Linear_combination
- In mathematics, a
**linear combination**is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of*x*and*y*would be any expression of the form*ax*+*by*, where*a*and*b*are constants).^{[1]}^{[2]}^{[3]}The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.Suppose that

*K*is a field (for example, the real numbers) and*V*is a vector space over*K*. As usual, we call elements of*V**vectors*and call elements of*K**scalars*. If*v*_{1},...,*v*_{n}are vectors and aa_{1},...,n are scalars, then the_{}*linear combination of those vectors with those scalars as coefficients*is [math]\displaystyle{ a_1 v_1 + a_2 v_2 + a_3 v_3 + \cdots + a_n v_n. \, }[/math]

- In mathematics, a

- ↑ Lay, David C. (2006).
*Linear Algebra and Its Applications*(3rd ed.). Addison–Wesley. ISBN 0-321-28713-4. - ↑ Strang, Gilbert (2006).
*Linear Algebra and Its Applications*(4th ed.). Brooks Cole. ISBN 0-03-010567-6. - ↑ Axler, Sheldon (2002).
*Linear Algebra Done Right*(2nd ed.). Springer. ISBN 0-387-98258-2.

### 2013

- http://en.wikipedia.org/wiki/Linear_combination#Functions
- Let
*K*be the set**C**of all complex numbers, and let*V*be the set C_{C}(*R*) of all continuous functions from the real line**R**to the complex plane**C**. Consider the vectors (functions)*f*and*g*defined by f*(*t*) :=*eit^{}*and*g*(*t*) :=*eit^{−}*. (Here,*e*is the base of the natural logarithm, about 2.71828..., and*i*is the imaginary unit, a square root of −1.) Some linear combinations of*f*and*g are:- [math]\displaystyle{ \cos t = \begin{matrix}\frac12\end{matrix} e^{i t} + \begin{matrix}\frac12\end{matrix} e^{-i t} \, }[/math]
- [math]\displaystyle{ 2 \sin t = (-i ) e^{i t} + (i ) e^{-i t}. \, }[/math]

- On the other hand, the constant function 3 is
*not*a linear combination of*f*and*g*. To see this, suppose that 3 could be written as a linear combination of*e*^{it}and*e*^{−it}. This means that there would exist complex scalars*a*and*b*such that*ae*^{it}+*be*^{−it}= 3 for all real numbers*t*. Setting*t*= 0 and*t*= π gives the equations*a*+*b*= 3 and*a*+*b*= −3, and clearly this cannot happen. See Euler's identity.

- Let