Linear Algebra Linear Independence Find all real numbers k s

Linear Algebra: Linear Independence

Find all real numbers k such that the polynomials 1 + kx + kx^2, k -kx + kx^2, k + kx +x^2 are linearly independent.

Solution

1 + kx + kx² = 0............. (1)

k + x + kx² = 0.......... (2)

k + kx + x² = 0 ............... (3)

(1) - (2)

(1-k) - x(1-k) =0

(1-x) (1-k) =0

k=1.