Question

Determine all values of the constant α for which {1 + αx2, 2 + x + x2, 3 + x} is a basis for p2. (enter your answers as a comma-separated list.)

Answer 1

The vectors is P_2 can be denoted as P_0 = 1 + ax^2, P_1 = 1 + x + x^2, and P_2 = 2 + x .

Determine W[P_0, P_1, P_2].


= (1+ax^2&a+x+x^2&2+x@2ax&1+2x&1@2a&2&0)
= (1 + ax^2) (0 - 2) – (1 + x + x^2) (0 – 2a) + (2 + x) (4ax – 2a – 4ax)
= -2 – 2ax^2 + 2a + 2ax + 2ax^2 – 4a – 2ax
= -2 -2a
If the given set is a basis for P_2, then they must be linearly independent. We can say that {P_0, P_1, P_2} is linearly dependent on any interval if determinant is not zero. Find the values of a for which the determinant is not equal to 0.
-2 – 2a != 0
-2 != 2a
-1 != a
Thus, the given set is a basis for P_2 for all a != -1