Working within the vector space P3 of polynomials of degree 3 or less

The question is if p can be written as a linear combination of the vectors in W . To check this, we set p equal to a linear combination and massage with the definitions of vector addition and scalar multiplication that we get with P₃ (Example VSP)

p(x) = a₁(x³ + x² + x) + a₂(x³ + 2x − 6) + a₃(x² − 5)

x³ + 6x + 4 = (a₁ + a₂)x³ + (a₁ + a₃)x² + (a₁ + 2a₂)x + (−6a₂ − 5a₃)

Equating coefficients of equal powers of x, we get the system of equations,

a₁ + a₂ = 1

a₁ + a₃ = 0

a₁ + 2a₂ = 6

−6a₂ − 5a₃ = 4

The augmented matrix of this system of equations row-reduces to

Since the last column is a pivot column, Theorem RCLS implies that the system is inconsistent. So there is no way for p to gain membership in W , so p 6∈ W .