# A problem on finding orthogonal basis and projection

In summary, for a continuous function on the interval [0, 1], we use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 to find an orthogonal basis for span = {x, x^2, x^3}. To project the function y = 3(x+x^2) onto this basis, we can use the Gram-Schmidt orthogonalization algorithm or make educated guesses for a suitable basis.
Use the inner product <f,g> = integral f(x) g(x) dx from 0 to 1 for continuous functions on the inerval [0, 1]

a) Find an orthogonal basis for span = {x, x^2, x^3}

b) Project the function y = 3(x+x^2) onto this basis.
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I know the following:
Two vectors are orthogonal if their inner product = 0
A set of vectors is orthogonal if <v1,v2> = 0 where v1 and v2 are members of the set and v1 is not equal to v2
If S = {v1, v2, ..., vn} is a basis for inner product space and S is also an orthogonal set, then S is an orthogonal basis.

Regarding projection, I know that if W is a finite dimensional subspace of an inner product space V and W has an orthogonal basis S = {v1, v2, ..., vn} and that u is any vector in V then,
projection of u onto W = <u, v1> v1/||v1||^2 + <u, v2> v2/||v2||^2 + <u, v3> v3/||v3||^2 + ...<u, vn> vn/||vn||^2

I can calculate integrals, but I really do not know how to fit all these together for this problem. I am not sure how to start.

Either you guess a basis which seems possible with a few try and error attempts, or you formally apply the Gram-Schmidt orthogonalization algorithm. If you have the new basis ##\{\,f(x),g(x),h(x)\,\}## then write ##3x+3x^2 = \alpha_1f(x)+\alpha_2g(x)+\alpha_3h(x)\,.##

## 1. What is an orthogonal basis?

An orthogonal basis is a set of vectors that are all mutually perpendicular to each other. This means that the dot product of any two vectors in the basis is equal to zero. Having an orthogonal basis is important in many mathematical and scientific applications, as it simplifies calculations and makes certain operations easier to perform.

## 2. Why is finding an orthogonal basis important?

Finding an orthogonal basis is important because it allows us to express a vector in terms of a set of basis vectors that are mutually perpendicular. This can simplify many calculations, such as finding the length of a vector or projecting a vector onto another vector. It also allows us to easily find the components of a vector in different directions.

## 3. What is the process for finding an orthogonal basis?

The process for finding an orthogonal basis involves first selecting a set of linearly independent vectors. Then, we use the Gram-Schmidt process to orthogonalize these vectors. This process involves subtracting the projection of each vector onto the previous vectors in the basis, until all vectors are mutually perpendicular. Finally, we normalize the vectors to have a length of 1, creating an orthonormal basis.

## 4. How is an orthogonal basis used in projection?

An orthogonal basis is used in projection by allowing us to find the component of a vector in a specific direction. This is done by taking the dot product of the vector with each basis vector. The resulting values represent the lengths of the projections of the vector onto each basis vector. By adding these projections together, we can find the total projection of the vector onto the subspace spanned by the basis vectors.

## 5. In what situations would finding an orthogonal basis be useful?

Finding an orthogonal basis is useful in many situations, including linear algebra, signal processing, and computer graphics. It is also used in various scientific fields, such as physics, engineering, and statistics. Anytime we need to simplify calculations involving vectors or find the components of a vector in different directions, an orthogonal basis can be very helpful.

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