Email: support@essaywriterpros.com
Call Us: US - +1 845 478 5244 | UK - +44 20 7193 7850 | AUS - +61 2 8005 4826

Polynomials

Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalizations below). However, the set S that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors. Also, there is no reason that n cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V.

Examples and counterexamples[edit]

This section includes a list of references, related reading or external linksbut its sources remain unclear because it lacks inline citations. Please help to improve this section by introducing more precise citations. (August 2013) (Learn how and when to remove this template message)

Euclidean vectors[edit]

Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R3. Consider the vectors e1 = (1,0,0), e2 = (0,1,0) and e3 = (0,0,1). Then any vector in R3 is a linear combination of e1e2 and e3.

To see that this is so, take an arbitrary vector (a1,a2,a3) in R3, and write:{\displaystyle (a_{1},a_{2},a_{3})=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\,}

(a_{1},a_{2},a_{3})=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\,

{\displaystyle =a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\,}

=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\,

{\displaystyle =a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}.\,}

=a_{1}e_{1}+a_{2}e_{2}+a_{3}e_{3}.\,

Functions[edit]

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 gdefined 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:

  • {\displaystyle \cos t={\begin{matrix}{\frac {1}{2}}\end{matrix}}e^{it}+{\begin{matrix}{\frac {1}{2}}\end{matrix}}e^{-it}\,}
  • {\displaystyle 2\sin t=(-i)e^{it}+(i)e^{-it}.\,}

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 eit. This means that there would exist complex scalars a and b such that aeit + beit = 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.