Determine whether the given set is a basis for the vector sp

Determine whether the given set is a basis for the vector space and refers to.

Determine whether the set of vector is linearly dependant or independent En P2\'. x^2 - 1, x^2 - 2, x^2 - 3

Solution

Basis B for a vector space V is a linearly independent subset of V that generates V. We need to see if these vectors are linearly independent or not.

Now we can write the vcetors as linear combination: a(x^2-1) +b(x^2 -2) +c(x^2 -3) =0

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

a +b +c =0

a-2b -3c =0

multiply equation 1 by 3 and then add it to 2nd equation:

4a+b=0

b = -4a

So, c = -b -a = -b -(-4a) = 3a

So,  a(x^2-1) -4a(x^2 -2) +3a(x^2 -3) =0

(x^2 -1) + (x^2 -3) = 4(x^2 -2)

The vectors are linearly dependent as we can express v1+v3 = 4v2