- Let $\beta = \{1 − x, 2x + x^2 , 3 + 2x^2 \}$ be a basis of $P_2$ (the vector space of the polynomials of degree less than or equal to 2.
In $P_2$ consider the inner product $< u, v >= a_0b_0 + a_1b_1 + a_2b_2$
where $u = a_0 + a_1x + a_2x^2$ and $v = b_0 + b_1x + b_2x^2$ are two vectors from $P_2$.
a) Verify that the basis $\beta$ is orthogonal to this inner product.
b) Find an orthogonal basis for $P_2$, according to the inner product presented.
c) Find an orthonormal basis for $P_2$, according to the inner product presented.
By calculations we obtain: $$<1-x,2x+x^2>=-2\ne 0\\<3+2x^2,2x+x^2>=2\ne 0\\<1-x,3+2x^2>=3\ne 0$$which is not orthogonal.
A basis which is both orthogonal and orthonormal is $\{1,x,x^2\}$