Redo Exercise 4.4.12 using the weighted inner product ⟨𝐯, 𝐰⟩=v1 w1+2 v2 w2+3 v3 w3 instead of th

Redo Exercise 4.4.12 using the weighted inner product ⟨𝐯,...

Key Concepts

Inner Product Space

An inner product space is a vector space equipped with an inner product, which is a function that assigns a scalar to each pair of vectors and satisfies properties such as linearity, symmetry (or conjugate symmetry in the complex case), and positive definiteness. This structure enables definitions of geometric concepts like length (using the induced norm) and angles between vectors.

Weighted Inner Product

A weighted inner product is a modification of the standard inner product where each component of the vectors is multiplied by a specified weight before summing. In this setting, different coordinates contribute unequally to the inner product, altering the geometry of the space. This concept is essential in applications requiring emphasis on certain dimensions over others, and it leads to a modified norm and notion of orthogonality.

Orthogonality

Orthogonality in an inner product space refers to two vectors being perpendicular if their inner product is zero. This concept extends naturally in spaces with different inner products, including those with weights, and is crucial for decomposing vectors into independent components, as well as for constructing orthogonal (or orthonormal) bases.

Projection

Projection involves finding the component of one vector along the direction of another or onto a subspace, such that the difference (the error) is orthogonal to the subspace. In a weighted inner product space, the projection formula is adapted by using the weighted inner product, which affects the way the scalar multiple is computed to account for the different weights assigned to each coordinate.

Transcript

00:01 Hello, for the following exercise we got these vectors here and a scalar, defined in this way.

00:11 So now let's consider the following inner product.

00:15 So let's suppose that we got this inner product defined technically is a weighted euclidean inner product.

00:23 So by weight means that we are given more importance and more weight to some of the components of the vectors and so that the the usual multiplication change.

00:38 So in this case, it's defined as follows.

00:41 Here is the first components of the vectors plus 5 u2v2.

00:48 So you can notice the difference with the usual equivalent in the product, which is defined just by the multiplication.

00:55 So in this case is called weight because we assign some weights to each of the components, so some of them have more like influence on the result.

01:05 So we have defined this and we need to make some compute some quantities.

01:12 So let's start.

01:14 So first let's start by computing this inner product of these two vectors.

01:21 So uv well is defined as one half and here we got the first component of the vector you and v is 1 times 3 plus 5 2 times 1 okay so at the end this is just 23 divided 2 okay so this is the result for the first part let's continue now we need to compute the inner product of the following quantity here we got k v and w so here k is just an scaler so we can take it out from the inner so at the end we obtain here k that multiplies the inner product of b with w.

02:08 This inner product here is equal to minus 10 and k is 3 so the result is minus 30.

02:18 For the next we got here a summation so we got u plus v w.

02:27 So we can solve this in two ways...

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