Show that S x2 3x 1 2x2 x 1 4x is or is not a basis for

Show that S = {x^2 + 3x + 1, 2x^2 + x - 1, 4x} is or is not a basis for P_2?

a( x^2 +3x -1) +b(2x^2 +x -1) +c(4x) =0

x^2(a +b) + x( 3a +b +4c) -a -b =0

a+b =0

3a +b +4c =0

-a -b =0

a = -b

So, 3a -a +4c =0 ; c= a

So,b = -a , c = a ;

So, a = b = c =0

Hence vectors are linearly indpenedent and S is a basis for P2

$Show that S = {x^2 + 3x + 1, 2x^2 + x - 1, 4x} is or is not a basis for P_2?Solutiona( x^2 +3x -1) +b(2x^2 +x -1) +c(4x) =0 x^2(a +b) + x( 3a +b +4c) -a -b =0$

Show that S x2 3x 1 2x2 x 1 4x is or is not a basis for

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