Linear Algebra

C22y In the vector space of polynomials P3, determine if the set S is linearly independent or linearly dependent. S={ { 2+x-3{ x }^{ 2 }-8 }{ x }^{ 3 },1+x+5{ x }^{ 3 },3-4{ x }^{ 2 }-7{ x }^{ 3 } }

\begin S={ { 2+x-3{ x }^{ 2 }-8 }{ x }^{ 3 },1+x+5{ x }^{ 3 },3-4{ x }^{ 2 }-7{ x }^{ 3 } } \end

\ S={ { v }{ 1 },{ 2 },_{ 3 } } \

S is linear independent if the equation { a }{ 1 }{ v }{ 1 }+{ a }{ 2 }{ v }{ 2}+{ a }{ 2 }{ v }{ 2 }=0 has the only trivial solution (0,0,0) with

{ a }{ 1 },{ 2 }{ ,a }_{ 3 }inR

Solving the equation { a }{ 1 }( { 2+x-3{ x }^{ 2 }-8 }{ x }^{ 3 } ) { +a }{ 2 }( 1+x+5{ x }^{ 3 } ) +_{ 3 }( 3-4{ x }^{ 2 }-7{ x }^{ 3 } ) =0

We have a system of 4 equations and 3 unknowns

\begin{cases} 2{ a }{ 1 }{ +a }{ 2 }+3{ a }{ 3 }=0 \ { a }{ 1 }{ +a }{ 2 }=0 \ -3{ a }{ 1 }+{ a }{ 2 }{ -4a }{ 3 }=0 \ -8{ a }{ 1 }+{ 5a }{ 2 }-7{ a }{ 3 }=0 \end{cases}\begin{cases} 2{ a }{ 1 }{ -a }{ 1 }+3{ a }{ 3 }=0 \ { a }{ 1 }{ =-a }{ 2 } \ -3{ a }{ 1 }{ -a }{ 1 }{ -4a }{ 3 }=0 \ -8{ a }{ 1 }{ -5a }{ 1 }-7{ a }{ 3 }=0 \end{cases}\begin{cases} { a }{ 1 }+3{ a }{ 3 }=0 \ { a }{ 1 }{ =-a }{ 2 } \ -4{ a }{ 1 }{ -4a }{ 3 }=0 \ -{ 13a }{ 1 }-7{ a }{ 3 }=0 \end{cases}\ \ \begin{cases} { a }{ 1 }+3{ a }{ 3 }=0 \ { a }{ 1 }{ =-a }{ 2 } \ { a }{ 3 }{ =-a }{ 1 } \ -{ 13a }{ 1 }-7{ a }{ 3 }=0 \end{cases}\begin{cases} { a }{ 1 }-3{ a }{ 1 }=0 \ { a }{ 1 }{ =-a }{ 2 } \ { a }{ 3 }{ =-a }{ 1 } \ -{ 13a }{ 1 }+7{ a }{ 1 }=0 \end{cases}\begin{cases} { a }{ 1 }=0 \ { a }{ 2 }{ =0 } \ { a }_{ 3 }{ =0 } \ \end{cases}

S is lineary independent.