Let R^3 have the inner product ⟨𝐮, 𝐯⟩=u1 v1+2 u2 v2+3 u3 v3 Use the Gram-Schmidt process to tran

Let R^3 have the inner product ⟨𝐮, 𝐯⟩=u1 v1+2 u2 v2+3 u3 v3...

Key Concepts

Inner Product Space

An inner product space is a vector space along with an additional structure called an inner product. This inner product is a mathematical tool that allows you to define geometric notions such as lengths (norms), angles, and orthogonality between vectors, providing a rigorous framework for many operations in linear algebra.

Custom Inner Product

A custom inner product is an inner product that is defined by a specific formula that might differ from the standard dot product. In this setting, coefficients or weights are incorporated into the product definition, which alters the interpretation of length and angle, and therefore changes notions like orthogonality and norm within the space.

Orthonormal Basis

An orthonormal basis is a set of vectors in an inner product space that are all mutually orthogonal and of unit length. Such bases are particularly valuable because they simplify vector decompositions and calculations of projections, and are foundational in many applications such as signal processing and numerical analysis.

Gram-Schmidt Process

The Gram-Schmidt process is a method for converting a set of linearly independent vectors into a set of orthonormal vectors that span the same subspace. The process works iteratively, subtracting components along previously determined basis vectors and then normalizing, and is applicable in any inner product space, including those defined with a custom inner product.

Transcript

0:00 Hello there.

00:01 So for this exercise we have a set of vectors you want you to you three and we need to apply the granite schmidt procedure to obtain a set of unitary orthogonal vectors v1 v2 b3 in this case i'm going to the node with the hat the unitary vector so if you see any vector that doesn't have the hat then is not unitary here are defined the vectors you want you two you three but here we have a different we have a particularity for this exercise and is that we're going to use the following definition of inner product that means if you can observe is a weighted inner product so it's a different way to obtain orthogonal vectors so it's not the same as considering the usual you clean an inner product.

00:58 So we need to be careful.

01:00 Always when you are asked to apply some gran schmidt procedure, always ask on which inner product space you're working or which what is the definition of the inner product in the space that you're working.

01:15 So having clarified that, let's start with the procedure.

01:19 That is the same algorithm.

01:20 The only thing that changed is how we're calculating the inner products.

01:25 So, we start by adding the first vector that will be orthogonal in our orthogonal set, so this will be an orthogonal set, and the first vector is always fixing some vector of your space that you're transforming into a set of our orthogonal vectors.

01:49 So just for to make it easier, always pick the first one that you have, or if you want to choose, for example, you two or three, it doesn't matter it's up to your choice so in this case i'm going to choose u1 so b1 will be u1 and this is equals to 1 1 1 then we need to add the second vector so the second vector we have b1 in our set in our orthogonal set and then we need to add a vector v2 so in the second step what we're going to do is transform our vector u2 into vector v2 such that v2 is orthogonal to v1.

02:36 So by doing this means that we're going to take v2 equal to u2 minus the projection of u2 on the over v1.

02:55 So this is the easiest way to obtain an orthogonal vector to v1.

03:00 And we know the definition of the projection is given by the inner product of u2 would v1 times v1 and divided the square of the norm of v1 so we have this in this case the inner product of u1 sorry u2 would v1 is equals you can observe is 1 plus one plus 2 times 1 plus 1 times 1 and 1 times 0 that is just 0.

03:50 So in this case this inner product with this definition of the inner product here is equal to 3...

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