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Mathematik-Online problems:

Problem 14: Vector Space Axioms and Linear (In)Dependence

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Let $ P_2(\mathbb{R})$ be the set of all real polynomials $ p$ with $ p'''=0$.
a)
Proof: $ P_2(\mathbb{R})$ is a vector space.
b)
Give a basis of $ P_2(\mathbb{R})$ (as simple as possible).
c)
Show that the polynomials $ p_1(x)=x^2+x+2$, $ p_2(x)=3x^2+2x+6$ and
$ p_3(x)=x-1$ are linearly independent.
d)
Describe the polynomial $ q(x)=x$ as linear combination of $ p_1$, $ p_2$ and $ p_3$.

(Authors: Kimmerle/Roggenkamp/Höfert)

see also:

[Solutions]
  automatisch erstellt am 14. 10. 2004