In the vector space of polynomials P3, determine if the set S is linearly independent or linearly dependent

Begin with a relation of linear dependence (Definition RLD),

a₁ (2 + x − 3x² − 8x³ )+ a₂ (1 + x + x² + 5x³ )+ a₃₍ 3 − 4x² − 7x³ )= 0

Massage according to the definitions of scalar multiplication and vector addition in the definition of P₃

(Example VSP) and use the zero vector for this vector space,

(2a₁ + a₂ + 3a₃) + (a₁ + a₂) x + (−3a₁ + a₂ − 4a₃) x² + (−8a₁ + 5a₂ − 7a₃) x³ = 0 + 0x + 0x² + 0x³

The definition of the equality of polynomials allows us to deduce the following four equations,

2a₁ + a₂ + 3a₃ = 0

a₁ + a₂ = 0

−3a₁ + a₂ − 4a₃ = 0

−8a₁ + 5a₂ − 7a₃ = 0

Row-reducing the coefficient matrix of this homogeneous system leads to the unique solution a₁ = a₂ = a₃ =0.  

So the only relation of linear dependence on S is the trivial one, and this is linear independence for S

(Definition LI).