Orthonormal Set spanning the subspace (polynomials)

[Cassi]

Homework Statement

In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let x_n(t) = tⁿ for n = 0, 1, 2,... Prove that the functions y₀(t) = 1, y₁(t) = sqrt(3)(2t-1), and y₂ = sqrt(5)(6t²-6t+1) form an orthonormal set spanning the same subspace as {x₀, x₁, x₂}.

Homework Equations

The Attempt at a Solution

I was attempting to use the Legendre Polynomials Rules to show that these polynomials form the basis but when I devise the Legendre Polynomials, they are different than those given.

 

[LCKurtz]

Homework Statement

In the linear space of all real polynomials with inner product (x, y) = integral (0 to 1)(x(t)y(t))dt, let x_n(t) = tⁿ for n = 0, 1, 2,... Prove that the functions y₀(t) = 1, y₁(t) = sqrt(3)(2t-1), and y₂ = sqrt(5)(6t²-6t+1) form an orthonormal set spanning the same subspace as {x₀, x₁, x₂}.

Homework Equations

The Attempt at a Solution

I was attempting to use the Legendre Polynomials Rules to show that these polynomials form the basis but when I devise the Legendre Polynomials, they are different than those given.

What's stopping you? Can you prove they are orthogonal? Orthonormal? Span the same space?

 

[vela]

What are the "Legendre Polynomials Rules"?

 

[HallsofIvy]

So $x_0= 1$, $x_1= t$, and $x_2= t^2$. What subspace do those span?

$y_0(t) = 1$, $y_1(t) = \sqrt{3}(2t-1)$, and $y_2 = \sqrt{5}(6t^2-6t+1)$. Show that these span the same subspace as the above. To show that they are orthonormal (which would also show that they are independent) you need to do 6 integrals.

For example, $\int_0^1 (y_0(t))^2 dt$ must be equal to 1 while $\int_0^1 y_0y_1 dt$ must be equal to 0.