Redo Example 4.43 with the dot product replaced by the weighted inner product ⟨𝐯, 𝐰⟩=v1 w1+2 v2

Redo Example 4.43 with the dot product replaced by the...

Key Concepts

Weighted Inner Product

A weighted inner product is a generalization of the standard dot product where each pair of corresponding components in the vectors is multiplied by a specific weight before summing. This allows for different importance or scaling of various components in computations and can be represented in matrix form using a symmetric positive definite matrix.

Inner Product

An inner product is a function that maps two vectors to a scalar, satisfying properties such as linearity in each argument, symmetry (or conjugate symmetry in complex spaces), and positive definiteness. It provides the foundation for defining lengths and angles in a vector space.

Norm Induced by an Inner Product

Every inner product defines a norm, which is the measure of the length of a vector obtained by taking the square root of the inner product of the vector with itself. This norm is essential in analyzing distances and convergence within the vector space.

Orthogonality

Orthogonality is the concept of two vectors being perpendicular, meaning their inner product equals zero. In spaces with a weighted inner product, the condition for orthogonality takes into account the weighting factors, which can alter the geometric relationships between vectors compared to the standard dot product.

Transcript

0:00 Hello there.

00:01 So in this exercise we need to consider the following linear transformation defined by the matrix a equals to 1 -1 minus 1 -1 and the vector x that is equals to the 1 -1 so in this exercise we need to find all the vectors v that satisfy the following wind condition that is that the inner product of x with v is equal to the inner product of the of the vector of the transformation of the vectors with respect to this linear transformation ta.

00:34 So for that, let's first calculate what is the inner product of x with v.

00:39 And remember that in this case, v is a generic vector in r2.

00:42 So we're going to write v as v1, v2.

00:47 So in that case, this inner product here is equal to just v1 plus v2.

00:58 And then we need to calculate the transformations of the vectors so in that case t a x will be multiplying the matrix a equals to 1 1 minus 1 1 1 with the vector 1 and we obtain the vector to 0 the same for the vector v and that is equal to the multiplication of the matrix 1 -1 minus 1 -1 with the vector v1 b2 and the result is the vector b1 plus v2 and v2 minus v1 okay so we got the expressions for t a x and tav so then let's write the inner product so is t a x t a it's going to be equals to the dot product of these two vectors b1 plus v2 v2 and the result is two times b1 plus v2 and now we need to find the condition that satisfy that both of these inner products are the same so that means that the inner product of x with v is equal to the inner product of tax, tav, is equivalent to satisfy that v1 plus v2 is equal to two times v1 plus v2.

02:45 And that translates to just one equation that is that v1 plus v2 should be equal to 0...

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