Question

determine if the set of polynomials {x^2-2x+1, 2x^2+3x-4, -x^2+x+5} is a linearly independent set in P_2. Is it a basis for P_2? Why or why not?

Answer 1

one way is to put
\[a(x^2-2x+1)+b( 2x^2+3x-4)+c( -x^2+x+5)=0\] and see if you can solve for \(a,b,c\)

Answer 2

this would mean
\[a+bb-c=0\]\[-2a+3b+c=0\]\[a-4b+5c=0\]

Answer 3

typo there, first one would be \(a+2b-c=0\)

Answer 4

they all equal 0 so they are linearly independent

Answer 5

is it a basis for P_2? why or why not?