Coefficient comparison

The coefficient comparison is a method from linear algebra in which the coefficients of two linear combinations of a linearly independent subset of a vector space are compared. A polynomial space is often used as a vector space with monomials as a linearly independent subset, for example in partial fraction decomposition . One uses the fact that two linear combinations of the same linearly independent subset are exactly equal if the corresponding coefficients are equal.

Polynomials

Two polynomials

${\displaystyle P(x)=a_{0}+a_{1}\cdot x+a_{2}\cdot x^{2}+\dotsb +a_{n}\cdot x^{n}}$

and

${\displaystyle Q(x)=b_{0}+b_{1}\cdot x+b_{2}\cdot x^{2}+\dotsb +b_{n}\cdot x^{n}}$

are equal if and only if their coefficients match:

${\displaystyle a_{0}=b_{0},a_{1}=b_{1},\dotsc ,a_{n}=b_{n}.}$

example

The two polynomials and are given. For which values ​​of and are the two polynomials equal? ${\displaystyle P(x)=a\cdot x+2a+b}$ ${\displaystyle Q(x)=2x+1}$ ${\displaystyle a}$ ${\displaystyle b}$

Must apply:

1. ${\displaystyle P(x)=Q(x)}$
2. ${\displaystyle ax+2a+b=2x+1}$

So it is compared:

1. ${\displaystyle ax=2x}$(Comparison of the coefficients of ) ${\displaystyle x^{1}}$
2. ${\displaystyle 2a+b=1}$(Comparison of the coefficients of ) ${\displaystyle x^{0}}$

Solution: and ${\displaystyle a=2}$ ${\displaystyle b=-3}$

Trigonometric polynomials

${\displaystyle (-a-8b)\cdot \sin(2x)+(-b+8a)\cdot \cos(2x)=130\cdot \sin(2x)}$
The following are compared:

1. ${\displaystyle -a-8b=130}$(Comparison of the coefficients of ) ${\displaystyle \sin(2x)}$
2. ${\displaystyle -b+8a=0}$(Comparison of the coefficients of ) ${\displaystyle \cos(2x)}$

Solution: ; ${\displaystyle a=-2}$ ${\displaystyle b=-16}$

See also

• Identity theorem for power series

literature

• Guido Walz (Ed.): Lexicon of Mathematics . tape 3 (Inp-Mon). Springer Spektrum Verlag, Mannheim 2017, ISBN 978-3-662-53501-1 , p. 131 , doi : 10.1007 / 978-3-662-53502-8 .