Determine whether the set of vectors is linearly dependent o

Determine whether the set of vectors is linearly dependent or independent

Determine whether the set of vector is linearly dependant or independant En P_2: -x, x^2 - 2x, 3.x + 5x^2

P2 = -x, x^2 -2x , 3x+5x^2

Let the vectors be expressed as linear combination :

ay1 +by2+cy3 where y1, y2.y3 are the three vectors

a(-x) +b(x^2 -2x) +c(3x+5x^2) = 0

x^2(b +5c) +x(-a -2b +3c) =0

b +5c =0

Now b = -5c

-a -2b +3c =0

-a -2(-5c) +3c =0

a = 7c

So, a(-x) +b(x^2 -2x) +c(3x+5x^2) = 0

7(-x) -5(x^2 -2x) +(3x +5x^2) =0

So, (3x +5x^2) = 7x +5(x^2 -2x)

So, y3 = 7y1+5y2

Hence vectors are linearly dependent