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 v1,...,vn are vectors and a1,...,an 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 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.
- ↑ 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 CC(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) := e−it. (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 eit and e−it. This means that there would exist complex scalars a and b such that aeit + 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 K be the set C of all complex numbers, and let V be the set CC(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) := e−it. (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: