IGNOU MTE 2 2018 - Let P3 be the inner product space of polynomials of degree at most 3 o

Question

Let P3 be the inner product space of polynomials of degree at most 3 over R with respect to the inner product

{f,g} = ₀∫¹ f (x) g(x)dx.

Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of P3 generated by the vectors

{1 − 2x, 2x + 6x², −3x² + 4x³ }.

02 Feb 2018

Answer :

Word Count : 582

To find an orthonormal basis for the subspace of \( P_3 \) generated by the set of polynomials \[ \{ 1 - 2x, \, 2x + 6x^2, \, -3x^2 + 4x^3 \}, \] we will apply the Gram-Schmidt process. Here's a step-by-step outline of the process: ### Step 1: Start with the given set of polynomials. Let the polynomials be: \[ v_1 = 1 - 2x, \quad v_2 = 2x + 6x^2, \quad v_3 = -3x^2 + 4x^3. \] ### Step 2: Orthogonalize the set. We begin by setting the first vector \( u_1 = v_1 \), because there is no previous vector to project onto. #### First step: We just normalize \( u_1 \): \[ u_1 = v_1 = 1 - 2x. \] Now, we compute the norm of \( u_1 \): \[ \|u_1\|^2 = \langle u_1, u_1 \rangle = \int_0^1 (1 - 2x)^2 \, dx. \] Expanding \( (1 - 2x)^2 \) gives: \[ (1 - 2x)^2 = 1 - 4x + 4x^2. \] Now, we compute the integral: \[ \int_0^1 (1 - 4x + 4x^2) \, dx = \left[ x - 2x^2 + \frac{4x^3}{3} \right]_0^1 = 1 - 2 + \frac{4}{3} = \frac{1}{3}. \] Thus, \[ \|u_1\| = \sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}. \] Therefore, the normalized vector is: \[ e_1 = \frac{1 -

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