For what values of a, b are the vectors (1,1, a)^T and (b,-1,1)^T orthogonal (a) with respect to

For what values of a, b are the vectors (1,1, a)^T and...

Key Concepts

Weighted Inner Product

A weighted inner product is a modification of the standard inner product in which certain components of the vectors are scaled by specified weights. This type of inner product still satisfies the properties of linearity, symmetry, and positive-definiteness. Orthogonality under a weighted inner product is defined by adjusting the calculation to account for these weights, leading to different conditions for when two vectors are considered perpendicular.

Dot Product

The dot product is a standard inner product in Euclidean space where the product of two vectors is obtained by summing the products of their corresponding entries. Two vectors are orthogonal with respect to this product if and only if their dot product is zero. This concept is foundational in determining angles between vectors and forms the basis for many methods in linear algebra.

Inner Product Spaces

An inner product space is a vector space on which an inner product is defined, providing a way to measure angles and lengths. In these spaces, the concept of orthogonality – meaning the inner product of two vectors is zero – is central to many theoretical and practical applications in mathematics, physics, and engineering.

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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