A First Course in Linear Algebra · Problem C40

Determinants — Robert Beezer

Mathematics — Linear Algebra · linear-dependence

Problem

In Example LDP4 we determined that the set of five polynomials,

T

, is linearly dependent by a simple invocation of Theorem SSLD. Prove that

T

is linearly dependent from scratch, beginning with Definition LI. The set is

T = \{p_1, p_2, p_3, p_4, p_5\} \subseteq W

where

\begin{aligned} p_1 &= x^4 - 2x^3 + 2x^2 - 8x + 8\\ p_2 &= -x^3 + 6x^2 - 5x - 6\\ p_3 &= 2x^4 - 5x^3 + 5x^2 - 7x + 2\\ p_4 &= -x^4 + 4x^3 - 7x^2 + 6x\\ p_5 &= 4x^3 - 9x^2 + 5x - 6 \end{aligned}

and

W = \{p(x)\mid p \in P_4,\, p(2) = 0\}