Q Let P3 be the inner product space of polynomials of degree at most 3 over R with respect to the inner product f,g = lim1-0 integrate 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 6x^2,−3x^2 4x^3} .?

Question

Q Let P3 be the inner product space of polynomials of degree at most 3 over R with respect to the inner product f,g = lim1-0 integrate 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 6x^2,−3x^2  4x^3} .?

Accepted Answer

Ans.

Method to Solve :

Let
$v_{1} = 1$

Answer

Gram-Schmidt Orthogonalization Process

The Gram-Schmidt orthogonalization process is a method used to find an orthonormal basis for a given subspace of an inner product space. The process involves iteratively constructing orthogonal vectors and normalizing them to obtain an orthonormal basis.

Step 1: Finding the First Vector

To begin the process, we will find the first vector in the orthonormal basis. In this case, the given vectors are {1 - 2x, 2x, 6x^2, -3x^2 + 4x^3}. We can choose any non-zero vector from this set as the first vector. Let's choose the first vector as v1 = 1 - 2x.

Step 2: Orthogonalizing the Second Vector

Next, we need to orthogonalize the second vector, v2 = 2x, with respect to v1. We can do this by subtracting the projection of v2 onto v1 from v2. The projection of v2 onto v1 is given by:

proj(v2, v1) = (v2 · v1) / ||v1||^2 * v1

where · represents the inner product and ||v1|| is the norm of v1. In this case, the inner product is defined as f, g = lim1-0 integrate f(x)g(x)dx, and ||v1||^2 = (v1 · v1).

Calculating the projection, we have:

proj(v2, v1) = ((2x) · (1 - 2x)) / ((1 - 2x) · (1 - 2x)) * (1 - 2x)
= (2x - 4x^2) / (1 - 4x + 4x^2) * (1 - 2x)
= (2x - 4x^2) / (1 - 4x + 4x^2 - 2 + 4x)
= (2x - 4x^2) / (3 - 4x + 4x^2)

Subtracting the projection from v2, we get:

u2 = v2 - proj(v2, v1)
= 2x - (2x - 4x^2) / (3 - 4x + 4x^2)

Simplifying the expression, we have:

u2 = (6x^2 - 3x^2 + 4x^3) / (3 - 4x + 4x^2)
= (3x^2 + 4x^3) / (3 - 4x + 4x^2)

Step 3: Orthogonalizing the Third Vector

Now we need to orthogonalize the third vector, v3 = 6x^2, with respect to v1 and v2. Following the same procedure as before, we calculate the projection of v3 onto v1 and v2, and subtract them from v3.

proj(v3, v1) = ((6x^2) · (1 - 2x)) / ((1 - 2x) · (1 - 2x))

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